**Author**- Zeldovich BYa. Perelomov AM.
**Popov**VS. **Title**- Relaxation of a quantum oscillator.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.55, no.2, 1968, pp.589-606. USSR.
**Abstract**- Relaxation of a quantum harmonic oscillator to the thermodynamic equilibrium state is considered on basis of a kinetic equation. The method of generating functions is employed for solving the equation. A number of exact solutions for various initial conditions is considered. For large quantum numbers it can be transformed to the Fokker-Planck equation. The Green function for the latter is determined. It is shown how the results obtained can be applied to the problem of statistics of photocounts. The given kinetic equation is extended to the case when an external classical force f(t) which arbitrarily depends on time acts on the oscillator. Methods for describing the evolution of the oscillator (characteristic functions, distributions of quasiprobabilities on the alpha plane) are briefly considered. The methods are valid in the presence of a force f(t). The equations of motion of the corresponding quantities are derived. The partial solutions for the density matrix rho are presented in a table. The so-called 'harmonic oscillator paradox' is discussed.

**Author**- Kotova LP. Perelomov AM.
**Popov**VS. **Title**- Quasiclassical approximation in ionization problems.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.54, no.4, April 1968, pp.1151-61. USSR.
**Abstract**- The paper is dedicated to the development of the complex trajectory technique (description of the tunnel effect in the quasiclassical case with the help of trajectories corresponding to imaginary 'times'). The following problems are considered: ionization of a bound level in collisions with a charged particle; ionization of atoms by a constant electric field in the presence of a magnetic field; evaluation of adiabatic corrections to the probability for tunnelling through a potential barrier of arbitrary shape.

**Author**- Belavin AA. Zel'dovich BYa. Perelomov AM.
**Popov**VS. **Title**- Relaxation of quantum systems with equidistant spectra.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.56, no.1, 1969, pp.264-74. USSR.
**Abstract**- The structure of a kinetic equation which describes the relaxation of quantum systems with equidistant spectra is discussed. Examples of such systems are harmonic oscillators (with linear or nonlinear damping), a spin in a magnetic field, an ensemble of two-level atoms interacting with a radiation field (Dicke problem). It is demonstrated that equations of this type retain positive determinacy of the density matrix. A solution of the kinetic equation is obtained for the Dicke problem and an exact quantum expression is found for the shape of the radiation line. Interference line narrowing due to the equidistant nature of the spectrum is discussed.

**Author****Popov**VS. Perelomov AM.**Title**- Parametric excitation of a quantum oscillator.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.56, no.4, 1969, pp.1375-90. USSR.
**Abstract**- An exact solution is obtained for the problem of a quantum oscillator with a time dependent frequency omega (t) varying in an arbitrary manner. The quantum transition probabilities w/sub mn/ between the discrete spectrum states which are stationary for t to +or- infinity are calculated. The variation of the adiabatic invariant I=E/ omega under the action of a variable frequency omega (t) is calculated for an arbitrary initial state.

**Author**- Zel'dovich BYa. Perelomov AM.
**Popov**VS. **Title**- Relaxation of quantum oscillator in the presence of an external force.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.57, no.1, 1969, pp.196-206. USSR.
**Abstract**- A solution of the kinetic equation for a quantum oscillator with damping on which an external classical force f(t) is acting has been obtained. A number of exact solutions for various initial conditions are considered and the spectrum of the radiation emitted by the oscillator is determined.

**Author****Popov**VS. Perelomov AM.**Title**- Parametric excitation of a quantum oscillator. II.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.57, no.5, Nov. 1969, pp.1684-90. USSR.
**Abstract**- For Pt.I see ibid. 56, 1375, 1969. An exact solution of the Schroedinger equation is obtained for a quantum oscillator with a varying frequency omega (t) which is acted upon by an external force f(t). The time dependence of omega (t) and f(t) may be arbitrary. The population distribution at the end of the processes is calculated for the case when at t to - infinity the oscillator was in the ground state mod 0, omega /sub -/). The quasi-energy spectrum for an oscillator with periodically varying parameters is determined.

**Author****Popov**VS.**Title**- On the mean diffusion escape time.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.58, no.4, 1970, pp.1400-5. USSR.
**Abstract**- The problem of determining the mean time for escape from an arbitrary region by particles performing Brownian movement or diffusion is considered. The dependence of the diffusion coefficient on coordinates is assumed to be arbitrary. In the case of spherical symmetry the expressions for the mean escape time tau and its dispersion Delta tau /sup 2/ can be reduced to quadratures.

**Author**- Gribov VN. Kobsarev IYu. Mur VD. Okun LB.
**Popov**VS. **Institution**- Acad. Scis., Moscow, USSR.
**Title**- On properties of amplitudes not satisfying the Pomeranchuk theorem conditions.
**Source**- Physics Letters B, vol.32b, no.2, 8 June 1970, pp.129-32. Netherlands.
**Abstract**- Properties of the scattering amplitudes as functions of the momentum transfer (t) and of the impact parameter ( rho ) are investigated in the case when the total cross sections for particle (p) and antiparticle (a) are symptotically (s to infinity ) constant and are not equal to each other. It is shown that if the equality Imf/sup p/(s,t)=Imf/sup a/(s,t) is violated, the weaker integral equality integral ImF/sup p/(s,t)s/sup -2/ds approximately= integral ImF/sup a/(s,t)s/sup -2/ds, t[left angle bracket]0 must hold. The equality provides for the absence of singularities at t=0 in the amplitude. It follows from this equality that t-(and s-) dependence of ImF/sup p/ and ImF/sup a/ must be different.

**Author****Popov**VS.**Institution**- USSR Acad. Sci., USSR.
**Title**- Electron energy levels at Z>137.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.11, no.5, 5 March 1970, pp.254-6. USSR.
**Abstract**- As is well known, the solution of the Dirac equation in the field of a point charge Ze is mathematically correct only if Z[left angle bracket]137. The energy of the lower level of the discrete spectrum (n=1, j=1/2) is epsilon /sub 0/= square root (1-a/sup 2/) (a=Ze/sup 2/=Z/137, h(cross)=c=m=1). (1) epsilon /sub 0/ reaches zero when alpha =1, and the continuation of formula (1) to the region alpha [right angle bracket]1 leads to imaginary values of epsilon /sub 0/. When alpha [right angle bracket]1 it is therefore necessary to take into account the finite dimensions of the nucleus. A critical charge is reached which determines the 'electrodynamic boundary' of the periodic system of elements and is also the pair production threshold.

**Author****Popov**VS.**Title**- 'Fall down to the center' at Z>137 and critical nuclear charge.
**Source**- Yadernaya Fizika, vol.12, no.2, 1970, pp.429-47. USSR.
**Abstract**- Solution is considered of the Dirac equation for electron in the Coulomb field with Z[right angle bracket]137, when for the point nuclear charge an 'incidence to the center' occurs. Account of a finite size of the nucleus resulting in regularization of the Coulomb potential at r to 0 allows one to deal with charges Z[right angle bracket]137 up to a 'critical' value Z/sub cr/, such that the ground state energy reaches the boundary of the lower continuum, E=-mc/sup 2/. An equation for Z/sub cr/ is obtained and dependence of Z/sub cr/ on the nuclear radius R and on the charge distribution inside the nucleus is studied. Mean radius of the ground state is calculated and it is shown that at Z to Z/sub cr/ the electron state remains localized, though Sigma =E/mc/sup 2/ to -1 and lambda = square root (1- epsilon /sup 2/) to 0. Relativistic Coulomb problem is also discussed for particles with spins 0 and 1.

**Author****Popov**VS.**Title**- Positron production in a coulomb field with Z>137.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.59, no.3, 1970, pp.965-84. USSR.
**Abstract**- Spontaneous production of positrons in a strong coulomb field with Z[right angle bracket]Z/sub cr/ is considered, where Z/sub cr/ is the critical value of the nuclear charge (for Z=Z/sub cr/ the energy of /sup 1/S/sub 1/2/ ground state reaches the limit of the lower continuum epsilon =-mc/sup 2/; the value of Z/sub cr/ always exceeds 137 and depends on the cut-off radius R and, to a less degree, on the nature of the electric charge distribution over the nuclear volume). It is shown that at the threshold (i.e. for (Z-Z/sub cr/)[left angle bracket][left angle bracket]Z/sub cr/) the probability for spontaneous positron creation vanishes exponentially. It is important for the analysis that for epsilon =-mc/sup 2/ (m is the electron mass). In this respect this case sharply differs from the usual nonrelativistic situation with epsilon to +mc/sup 2/.

**Author**- Gribov VN. Kobsarev IYu. Mur VD. Okun LB.
**Popov**VS. **Title**- On properties of the amplitudes not satisfying the conditions of the Pomeranchuk theorem.
**Source**- Yadernaya Fizika, vol.12, no.6, pp.1271-85. USSR.
**Abstract**- Properties of the scattering amplitudes as functions of the momentum transfer t and of the impact parameter rho are considered in case when the total cross sections for a particle and the antiparticle are asymptotically (s to infinity ) constant and not equal. A detailed consideration is performed for a so called scale-invariant case when the partial amplitudes ( up to a common factor) depend only on one variable x= rho /1n s.

**Author****Popov**VS.**Title**- On the properties of a discrete spectrum for values of Z close to 137.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.60, no.4, 1971, pp.1228-44. USSR.
**Abstract**- The nature of the discrete spectrum for an electron in the Coulomb field of a bare nucleus of charge close to 137 is investigated. In this region of Z values the level spectrum significantly depends on cut-off of the Coulomb potential at small distances. An equation is derived for the level energies epsilon /sub n/. In the logarithmic approximation (i.e. on the condition that L=In (h(cross)/mcR)[right angle bracket][right angle bracket]1, where m is the electron mass and R the nuclear radius) the equation can be solved explicitly and the simple expressions can be obtained for epsilon /sub n/.

**Author****Popov**VS.**Institution**- Acad. Sci., USSR.
**Title**- Production of e/sup +/e/sup -/ pairs in an alternating external field.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.13, no.5, 5 March 1971, pp.261-3. USSR.
**Abstract**- Interaction of an electromagnetic field with a vacuum of charged particles leads to the appearance of nonlinear additions to the Lagrangian L of the electromagnetic field. In particular, there appears an imaginary part Im L, which determines the probability w of pair production by the external field. In the case of fields E and H constant in time (but alternating in space) it is possible to solve this problem exactly. The author using Dirac's theory in a quasiclassical approach obtains probabilities for pair production in time-varying fields E(t).

**Author****Popov**VS.**Title**- On the problem of the critical charge of nucleus.
**Source**- Yadernaya Fizika, vol.14, no.2, 1971, pp.458-68. USSR.
**Abstract**- The critical charge Z/sub cr/ is by definition the value of the charge of nucleus (or of the system of nuclei) at which the energy of the ground state of the discrete spectrum decreases down to the boundary of the lower continuum E=-mc/sup 2/. To calculate Z/sub cr/, the method of the effective potential is developed and the variational principle is formulated. With their help one can obtain Z/sub cr/ for the system of two colliding nuclei.

**Author****Popov**VS.**Title**- Pair production in a varying external field (quasiclassical approximation).
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.61, no.4, 1971, pp.1334-51. USSR.
**Abstract**- A quasiclassical method is developed for calculating the probability for pair production in a varying electromagnetic field. The conditions of applicability of the method are: E[left angle bracket][left angle bracket]m/sup 2/c/sup 3//eh(cross), h(cross) omega [left angle bracket][left angle bracket]mc/sup 2/, where omega is the effective field variation frequency, E is the field amplitude and m mass of the particles produced by the field. Underlying the method is the concept of sub-barrier particle motion taking place along a classical trajectory but for an imaginary value of 'time'. The exact Green function for the Dirac equation in an external field is not required and the method can be applied to a very broad class of fields.

**Author**- Migdal AB. Perelomov AM.
**Popov**VS. **Title**- Analytical properties of wave functions at low energy.
**Source**- Yadernaya Fizika, vol.14, no.4, 1971, pp.874-85. USSR.
**Abstract**- Analytical properties of the continuum wave functions chi /sub k/(r) are investigated for the case when there is a level with energy near zero (the level may be real, virtual, or quasistationary). It is shown that at small distances, r[left angle bracket][left angle bracket]k/sup -1/, (k is the wave vector) the wave functions became factorizable: chi /sub k/(r) approximately=( Delta (k))/sup 1/2/ chi /sub 0/(r) while the factor Delta (k) changes sharply within the resonance width. Delta (k) is expressed in terms of the scattering amplitude at small k and is determined by the analyticity.

**Author****Popov**VS. Rozhdestvenskaya TI.**Title**- Critical charge in collisions of nuclei.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.14, no.4, 20 Aug. 1971, pp.267-70. USSR.
**Abstract**- In order to verify the theory of supercritical atoms a calculation is made for the collision of the two bare uranium nuclei and the probability of positron emission as a function of internuclear distance.

**Author**- Perelomov AM.
**Popov**VS. **Title**- Perturbation of a continuous spectrum by a near-lying level.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.61, no.5, 1971, pp.1743-55. USSR.
**Abstract**- The nature of the perturbation of continuous spectrum functions chi /sub k/(r) and the analytic properties of the S matrix in the region k to 0 in the presence of a level close to zero are considered. Formulas are obtained which express parameters of the low energy expansion of the S matrix in terms of the wave function chi /sub 0/(r) at the moment of formation of the level. Exact solutions of the problem are obtained for a number of potentials.

**Author****Popov**VS.**Title**- Method for a calculation of the nuclear critical charge.
**Source**- Yadernaya Fizika, vol.15, no.5, May 1972, pp.1069-81. USSR.
**Abstract**- A simple formula is given for the nuclear critical charge. The formula is asymptotically exact at the condition R[left angle bracket][left angle bracket]h(cross)/(m/sub e/c), where R is the nuclear radius. The dependence of the critical charge Z/sub c/ on R, on to character of the charge distribution inside the nucleus, etc., is investigated by the use of this formula. Perturbation theory for Z/sub c/ is considered.

**Author**- Marinov MS.
**Popov**VS. **Title**- Pair production in electromagnetic field (case of arbitrary spin).
**Source**- Yadernaya Fizika, vol.15, no.6, 1972, pp.1271-85. USSR.
**Abstract**- The quasiclassical method of 'imaginary time' is generalized for the case of a particle with spin. The production probability of pairs in strong electromagnetic field for particles with an arbitrary spin and magnetic moment is calculated.

**Author**- Migdal AB. Perelomov AM.
**Popov**VS. **Title**- Notes on analytic properties of wave functions at very low energy.
**Source**- Yadernaya Fizika, vol.16, no.1, 1972, pp.222-4. USSR.
**Abstract**- Proposes some additions and refinements to results obtained in studies of the behaviour of continuous spectral wave functions in the case where there is an almost-zero energy level in the system, and in the so-called 'narrow-pit' example. In regard to wave function factorization, a refinement is introduced for barrier permeability. Also an interpretation of the 'narrow pit' leads to expansion of the region of applicability of factorization.

**Author**- Zel'dovich YaB.
**Popov**VS. **Title**- Electronic structure of superheavy atoms.
**Source**- Uspekhi Fizicheskikh Nauk, USSR.
**Abstract**- The authors describe the status of the problem of the electron structure of superheavy atoms with nuclear charge Z[right angle bracket]Z/sub c/; here Z/sub c/ approximately=170 is the critical value of the nuclear charge, at which the energy of the ground state of the 1S/sub 1/2/ electron reaches the limit of the lower continuum of the solutions of the Dirac equation ( epsilon =-m/sub e/c/sup 2/). They discuss the dependence of Z/sub c/ on the nuclear radius R and on the character of the distribution of the electric charge inside the nucleus, and also the form of the wave functions at Z close to Z/sub c/. Owing to the Coulomb barrier, the state of the electron remains localized at Z[right angle bracket]Z/sub c/, in spite of the fact that its energy approaches the continuum. An analysis of the polarization of the vacuum in a strong Coulomb field shows that a bare nucleus with supercritical charge Z[right angle bracket]Z/sub c/ produces spontaneously two positrons and, in addition a charge density with a total of two units of negative charge in the vacuum.

**Author****Popov**VS. Marinov MS.**Title**- e/sup +/e/sup -/-pair production in variable electric field.
**Source**- Yadernaya Fizika, vol.16, no.4, 1972, pp.809-22. USSR.
**Abstract**- The calculation of probability w of production of fermion pairs in vacuum under the influence of variable electric field E(t) comes to a solution of the problem on parametric excitation of quantum oscillator. The cases, when external field E(t) preserves constant direction in space or is situated in a definite plane, are treated in detail. By the 'imaginary time' method a closed formula has been obtained for probability w in the case of the field with elliptical polarization.

**Author****Popov**VS.**Title**- Critical charge in the two-center problem.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.16, no.6, 20 Sept. 1972, pp.355-8. USSR.

**Author****Popov**VS.**Institution**- Inst. Theoretical & Experimental Phys., USSR.
**Title**- Pair production in a variable and homogeneous electric field as an oscillator problem.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.62, no.4, April 1972, pp.1248-62. USSR.
**Abstract**- The calculation of the probability w of pair production in a strong electric field E(t) which is homogeneous in space reduces to the problem of the parametric excitation of a quantum oscillator (or, what is equivalent, to the calculation of the reflection coefficient for a plane wave incident on a one-dimensional potential barrier). Because of this relation, the calculation of the pair-production probability does not require the determination of the exact solutions of the relativistic wave equations and the Green's function. In the particular case when E(t)=E(cosh omega t)/sup -2/ the probability w coincides with the coefficient of reflection from the Eckart potential. The relation between the exact formulas for w and the quasiclassical approximation (the 'imaginary time' method) is investigated in detail, and the limits of applicability of the quasiclassical approximation are indicated.

**Author****Popov**VS.**Title**- Critical charge for colliding nuclei.
**Source**- Yadernaya Fizika, vol.17, no.3, 1973, pp.621-33. USSR.
**Abstract**- Using the approximation (Z/sub i/+Z/sub 2/-Z/sub c/)[left angle bracket][left angle bracket]Z/sub c/, formulae are obtained for the critical charge in relativistic two-center problem. It allows one to find a critical distance R/sub c/ (Z/sub 1/, Z/sub 2/) between colliding nuclei at which the ground state energy of quasimolecule (Z/sub 1/, Z/sub 2/, e) reaches the boundary of the lower continuum and spontaneous production of positrons begins.

**Author**- Gershtein SS.
**Popov**VS. **Institution**- Inst. High-Energy Phys., Serpukhov, USSR.
**Title**- Spontaneous production of positrons in collisions of heavy nuclei.
**Source**- Lettere Al Nuovo Cimento, vol.6, ser.2, no.14, 7 April 1973, pp.593-6. Italy.
**Abstract**- The authors show that spontaneous production of positrons takes place even when only one of the nuclei is bare (and the second one has an ordinary electron shell structure). This fact provides an opportunity to carry out the experiment with a heavy target.

**Author**- Marinov MS.
**Popov**VS. **Title**- Effect of screening on the critical charge of a nucleus.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.17, no.9, 5 May 1973, pp.511-4. USSR.
**Abstract**- In view of the difficulty of calculating Z/sub c/ in the two-center problem the authors consider, in order to determine the scale of the screening correction, a model problem with a spherical superheavy nucleus.

**Author****Popov**VS. Mur VD.**Title**- Quasistationary levels in lower continuum.
**Source**- Yadernaya Fizika, vol.18, no.3, 1973, pp.684-98. USSR.
**Abstract**- Quasistationary levels in the lower continuum of solutions of the Dirac equation are considered. Depth of the potential V(r) is assumed to be sufficient ( mod V mod [right angle bracket]2mc/sup 2/) for the bound states to cross the boundary of the lower continuum epsilon =-mc/sup 2/ at V=V/sub c/. In so doing the bound state becomes a quasistationary state whose width gamma determines probability of spontaneous positron production in the static field. Behaviour of gamma at the threshold V to V/sub c/ and at V[right angle bracket][right angle bracket]V/sub c/ is investigated. It is shown that gamma is limited even at mod epsilon mod [right angle bracket][right angle bracket]mc/sup 2/, i.e. for a level sunken deeply into the lower continuum. For the Coulomb field gamma /mc/sup 2/ approximately exp(-2 pi Z alpha )[left angle bracket][left angle bracket]1. This result shows that the description of the K shell of a supercritical atom with the nuclear charge Z[right angle bracket]Z/sub c/, (see abstr. A67385 of 1972), is valid not only at epsilon to -mc/sup 2/, but also for deep levels. The angular momentum (j) dependence of gamma and analytical properties of the S matrix near V=V/sub c/ are studied in detail. In particular, the case of the p/sub 1/2/ level, having some peculiar properties, is considered, as well as the s levels of a scalar particle.

**Author****Popov**VS.**Institution**- Inst. Theoretical & Experimental Phys., Acad. Sci., USSR.
**Title**- Spontaneous positron production by the Coulomb field of a heavy nucleus.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.18, no.1, 5 July 1973, pp.53-8. USSR.
**Abstract**- The spontaneous production of positrons in a Coulomb field with Z[right angle bracket]Z/sub c/ (where Z/sub c/ approximately=170 is the critical charge of the nucleus) is of interest as an example of e/sup +/e/sup -/ pair production via an unusual static mechanism. The most realistic way of searching for this phenomenon is in collisions of heavy nuclei with Z/sub 1/+Z/sub 2/[right angle bracket]Z/sub c/, for example uranium or californium nuclei. It is not necessary that both colliding nuclei be 'bare' (i.e., have no electron shells). When a beam of bare nuclei Z/sub 1/ is incident on an ordinary heavy target Z/sub 2/, vacant places are produced in the K shell of the combined atom if Z/sub 1/[right angle bracket]Z/sub 2/. The cross section sigma for the production of e/sup +/ has in this case practically the same value as for bare nuclei.

**Author****Popov**VS.**Institution**- Inst. Theoretical & Experimental Phys., USSR.
**Title**- The 'imaginary-time' method in problems concerning the ionisation of atoms and pair production.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.63, no.5, Nov. 1972, pp.1586-98. USSR.
**Abstract**- The 'imaginary-time' method was originally proposed in order to solve the problem of the ionisation of non-relativistic bound systems of atoms and ions in the field of an intense light wave. The method is a generalisation of the quasiclassical WKB approximation to the case of time-dependent fields. It describes the tunnelling transition of an electron from a bound state to the continuum by using the classical equations of motion, but with an imaginary time. Such an approach has also been found useful in connection with investigations of ionisation in Coulomb collisions. It has also been established that the method is successful in calculating the probability for pair production from vacuum due to an external field, although the relation between these two problems has remained nuclear. It is shown that the ionisation of atoms and the production of e/sup +/e/sup -/ pairs from vacuum correspond to two limiting cases of the general relativistic problem of the ionisation of a level with arbitrary energy E due to the influence of an internal electric field. Advantages associated with such a generalisation are discussed. In particular, by going to the non-relativistic limit a closed expression can be obtained for the atomic ionisation probability omega in an arbitrary electric field-a new result. The findings are of interest in laser technology.

**Author****Popov**VS.**Title**- Spontaneous production of positrons at collisions of heavy nuclei (production cross section and the energy spectrum of e/sup +/).
**Source**- Yadernaya Fizika, vol.19, no.1, 1974, pp.155-68. USSR.
**Abstract**- The analytical solution of the two-centre Coulomb problem for the Dirac equation is obtained in the approximation (Z/sub 1/+Z/sub 2/-Z/sub c/)/Z/sub c/[left angle bracket][left angle bracket]1. Using the solution, the total cross section sigma of positron production in the quasistatic field of the colliding nuclei, the cross section d sigma /d Omega corresponding to the scattering of the nucleus under a fixed angle theta , and the energy spectrum of the produced positrons are calculated. Dependence of these quantities on the incident nucleus energy E is considered. This enables one to suggest the optimal energy region for the experiment. The similar problem of e/sup +/e/sup -/-pair production from vacuum in a variable electric field E(t) is discussed briefly.

**Author****Popov**VS.**Institution**- Inst. Theoretical & Experimental Phys., USSR.
**Title**- Resonant pair production in a strong electric field.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.18, no.7, 5 Oct. 1973, pp.435-9. USSR.
**Abstract**- Pair production anomalies connected with the periodicity of the field are investigated. An analogy is found between the problems of fermion pair production from vacuum and resonant excitation of atomic levels in a strong optical wave field. The analogy no longer holds if excitation of multilevel atomic systems is considered.

**Author****Popov**VS.**Title**- Method of imaginary time for periodical fields.
**Source**- Yadernaya Fizika, vol.19, no.5, 1974, pp.1140-56. USSR.
**Abstract**- The method of imaginary times is formulated for periodical fields, T(t+T)=F(t), where F(t) is an external electric field satisfying the conditions necessary for the quasiclassical approach to be consistent. Ionization of atoms by a strong light wave and pair creation from vacuum are considered as applications. Explicit expressions for the pair production probability w, applicable for any periodical field, are obtained in case of the linear polarization of the field F(t). The probability w has different forms for fermions and for bosons. Effect of the statistics of the created particles is most essential at p/sub ///=0, i.e. for particles produced in a direction transversal to the electric field. In this case boson pairs are created only if an even number of quanta is absorbed, n=2K, and fermion pairs are created only if an odd number is absorbed, n=2K+1. Properties of the exact solutions of the Klein-Gordan and Dirac's equations with a periodical field and the relation to the quasiclassical approach are discussed. Pair creation by an electric field with elliptic and circular polarizations is also considered by means of the method of imaginary times.

**Author**- Perelomov AM.
**Popov**VS. **Title**- Variational principle for critical charge.
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.14, no.1, Jan. 1973, pp.18-26. USSR.
**Abstract**- In order to find the critical charge Z/sub V/, the effective potential method is developed and a variational principle is formulated.

**Accession Number**- 000676438
**Author**- Marinov MS.
**Popov**VS. Stolin VL. **Title**- Variational calculation of the critical distance between nuclei.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.19, no.1, 5 Jan. 1974, pp.76-80. USSR.
**Abstract**- A variational calculation is made of the critical distance R/sub c/ between colliding nuclei, at which the fundamental term of the quasimolecule (Z,Z,e) drops to the limit of the lower continuum and spontaneous positron generation begins.

**Author**- Marinov MS.
**Popov**VS. **Title**- A variant of the Wentzel-Kramers-Brillouin method and calculation of the critical nuclear charge.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.67, no.4, Oct. 1974, pp.1250-62. USSR.
**Abstract**- A modification of the WKB method is proposed which consists in exclusion of centrifugal energy from the momentum p(r) and in a corresponding alteration of the constant gamma in the quantization condition gamma = gamma (l,a), where l is the orbital moment and V(r) approximately r/sup - alpha / for r to 0. The accuracy of the approximation and application of the WKB method for a strong coulomb field with a charge Z[right angle bracket]137 are considered in detail.

**Author**- Marinov MS.
**Popov**VS. **Institution**- Inst. Theoretical & Experimental Phys., USSR.
**Title**- On the production of e/sup +/e/sup -/ pairs from the vacuum under the influence of an electric field.
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.17, no.1, Oct. 1973, pp.34-46. USSR.
**Abstract**- A study is made of the problem of pair production of particles with spin 1/2 in an external homogeneous electric field E(t). The Heisenberg equations of motion are found for the operators of creation and annihilation, a solution for which has the form of a linear canonical transformation. A system of differential equations that determine the coefficients U(t) and V(t) of this transformation are obtained. The probabilities of pair creation are expressed in terms of the eigenvalues of the matrix VV/sup +/. The case of particles with anomalous magnetic moment is also considered.

**Author**- Marinov MS.
**Popov**VS. **Title**- On energy spectrum of positrons produced in nuclear interactions at R
**Source**- Yadernaya Fizika, vol.20, no.6, 1974, pp.1223-8. USSR.
**Abstract**- The energy spectrum positrons produced in the collision of heavy nuclei with charges Z/sub 1/+Z/sub 2/[right angle bracket]Z/sub c/ is calculated when the ground level of the discrete spectrum dives into the lower continuum. The spectrum positron has a sharp peak near the upper boundary T=T/sub max/.

**Author**- Marinov MS.
**Popov**VS. **Title**- Critical distance in heavy ion collisions.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.62, no.2, Feb. 1975, pp.421-31. USSR.
**Abstract**- A variational calculation is carried out for the critical distance R/sub cr/ between colliding ions. Values of R/sub cr/ are calculated in the 90[left angle bracket]or=Z[left angle bracket]or=100 region with allowance for finite size of the nuclei. In this case R/sub cr/ greatly exceeds the nuclear radius r/sub N/ (for example R/sub cr/ approximately=50F for uranium nuclei). The results can be employed for checking the predictions regarding the cross section for spontaneous, positron production in Coulomb collisions of heavy nuclei.

**Author**- Mur VD.
**Popov**VS. **Title**- Pole model of the scattering amplitude with a highest possible growth of the cross section.
**Source**- Yadernaya Fizika, vol.21, no.4, 1975, pp.868-77. USSR.
**Abstract**- According to the known Froissart inequality, the total cross section sigma /sub t/ at s to infinity cannot grow more rapidly than ln/sup 2/ s. The pole models of partial amplitudes of the cross-channel leading to the highest possible growth sigma /sub t/ approximately ln/sup 2/ s and consistent with the unitarity condition in the s channel are considered. It is shown that in the region of the parameter rho approximately ln s' an automodel regime is being settled, i.e. the partial amplitudes of the s channel asymptotically depend only upon the ratio rho /ln s. General features of the amplitudes as well as the cases sigma /sub t/ approximately ln s are briefly discussed.

**Author****Popov**VS.**Title**- Spontaneous positron production in collisions between heavy nuclei.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.65, no.1, July 1973, pp.35-53. USSR.
**Abstract**- It is shown that in collisions between a completely stripped (bare) nucleus (Z/sub 2/) and the atoms of a heavy target (Z/sub 1/), vacancies are formed in the K-shell of the united atom, providing Z/sub 2/[right angle bracket]or=Z/sub 1/. Therefore such collisions, as well as those between two bare nuclei, will result in spontaneous production of electron-positron pairs in a static Coulomb field of the colliding nuclei if the nuclei approach each other at a distance R[left angle bracket]R/sub c/ (R/sub c/ is a critical distance at which the ground level of the discrete spectrum becomes as low as the boundary of the lower continuum). In this case one or two electrons fill the vacancies in the K-shell of the united atom and the positrons escape to infinity. This circumstance should make feasible the experimental observation of spontaneous production of positrons in an ordinary heavy target by a bare nuclear beam. The kinematics of the process is considered and the threshold behavior of the positron production cross section and positron energy spectrum are considered.

**Author**- Marinov MS.
**Popov**VS. **Institution**- Inst. for Theoretical & Experimental Phys., Moscow, USSR.
**Title**- A modification of the Bohr-Sommerfeld quantization condition.
**Source**- Journal of Physics A-Mathematical & General, vol.8, no.10, Oct. 1975, pp.1575-85. UK.
**Abstract**- The Bohr-Sommerfeld quasiclassical quantization condition for a central potential is modified in such a way that the constant gamma becomes dependent on the angular momentum l and on the potential behaviour at small distances. This form of the quantization condition is especially simple in the case l=0. When applied to some particular potentials, taken as examples, the present approximation provides more accurate values of the level energies than the conventional form of the quantization condition. Application of the quasiclassical approach to singular potentials with 'collapse' and to the superstrong Coulomb field induced by a charge Z[right angle bracket]137 is also considered.

**Author**- Marinov MS.
**Popov**VS. **Title**- Phase shift technique for the Dirac equation and the calculation of the critical nuclear charge.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.65, no.6, Dec. 1973, pp.2141-54. USSR.
**Abstract**- A simple method for solving the Dirac and Schroedinger equations with an arbitrary potential V(r) is developed which is convenient for qualitative analysis of the problem and for numerical calculations. The idea of the method is that the equation under consideration is replaced by first order nonlinear equation for the phase function theta (r). Application of the method to the problem of appearance of a bound level and its application for calculation of the critical charge Z/sub c/ are considered. The effect of screening of the nuclear Coulomb field by the electron shell on Z/sub c/ is estimated. Variational calculations of Z/sub c/ for a system of colliding nuclei (relativistic two-center problem) are performed.

**Author**- Marinov MS.
**Popov**VS. Stolin VL. **Institution**- Inst. for Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Variational approach to the relativistic two-center problem: Critical internuclear distance.
**Source**- Journal of Computational Physics, vol.19, no.3, Nov. 1975, pp.241-56. USA.
**Abstract**- As shown recently, if the distance between two heavy nuclei becomes less than its critical value, R/sub cr/, their electric potential is strong enough to create electron-positron pairs from the vacuum. This process may occur when the ground level of the quasimolecule (Z,Z,e) is lower than -m/sub e/c/sup 2/. In order to find R/sub cr/ the Dirac equation with two Coulomb centres is considered. By means of a proper choice of coordinates and using the variational principle the problem is reduced to a set of ordinary linear differential equations with singular coefficients. To solve the boundary problem the Riccati equation for the logarithmic derivative matrix was considered. The computations are carried out and R/sub cr/ is found as a function of the nuclear charges. By virtue of the variational principle the obtained quantity is a sure lower limit for R/sub cr/.

**Author**- Marinov MS.
**Popov**VS. **Title**- Two-center problem for the Dirac equation.
**Source**- Yadernaya Fizika, vol.23, no.2, 1976, pp.479-87. USSR.
**Abstract**- The two-center problem for the Dirac equation is considered in view of an experiment on the spontaneous production of positrons in collisions of heavy nuclei. The positrons are created when the internuclear distance is less than its critical value R/sub cr/. The critical distance R/sub cr/ is calculated by means of the variational approach for all the nuclear charges Z[left angle bracket]137. It is shown that at Z to 137 the function R/sub cr/(Z) has a finite limit. Formulae necessary for numerical calculations of the critical distance by means of the variational approach are presented.

**Accession Number**- 000977788
**Author**- Migdal AB. Voskresenskii DN.
**Popov**VS. **Title**- On the distribution of vacuum charge near supercharged nuclei.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.24, no.3, 5 Aug. 1976, pp.186-8. USSR.
**Abstract**- A supercritical nucleus, surrounded by a vacuum shell, which remains around the nucleus after the departure of the positrons to infinity, is investigated. The behaviour of the potential and of the density of the electron cloud, near the edge of the supercharged nucleus, at Ze/sup 3/[right angle bracket][right angle bracket]1, is determined. The analysis is performed in the Thomas-Fermi approximation.

**Accession Number**- 000992915
**Author**- Eletskii VL.
**Popov**VS. **Title**- Critical charge for abnormal nuclei.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.24, no.4, 20 Aug. 1976, pp.253-6. USSR.
**Abstract**- Results are presented of analysis and calculation of the critical charge Z/sub c/ for abnormal nuclei, specifically, a relationship is obtained for Z/sub c/ as a function of density of the nuclear substance and Z/A ratio. The results indicate that Z/sub c/ falls with rising n/sub p/ and, in the range n/sub p/[right angle bracket][right angle bracket]n/sub p//sup (o)/, approaches 137 value (critical charge for a point nucleus).

**Author****Popov**VS. Eletskii VL. Mur VD.**Title**- Properties of deep-lying levels in a strong electrostatic field.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.71, no.3, Sept. 1976, pp.856-71. USSR.
**Abstract**- Properties of deep levels (real, virtual and quasistationary) lying near the boundary of the lower continuum are considered. The effective radius expansion is extended to the case of the Dirac equation. It is employed for investigating the level motion near the boundary epsilon =-mc/sup 2/ for various values of the moments j and l. In the limiting cases of R[left angle bracket][left angle bracket]h(cross)/mc and R[right angle bracket][right angle bracket]h(cross)/mc (R is the range of forces) the formulas can be simplified. The case of a wide potential well R[right angle bracket][right angle bracket]h(cross)/mc, in particular, can be reduced to the determination of the spectrum of bound and quasistationary states in the Schrodinger equation with a power potential. The asymptotic behavior of the critical nuclear charge Z/sub cr/ in the region R/sub N/[right angle bracket][right angle bracket]h(cross)/mc (R/sub N/ is the nuclear radius) is found and Z/sub cr/ is calculated for the muon for various distributions of electric charge in the nucleus. Differences in the behavior of the levels near epsilon =-mc/sup 2/ for scalar and spinor particles and the inapplicability of the Klein-Gordon single particle equation for Z[right angle bracket]or=Z/sub cr/ are discussed.

**Author**- Migdal AB.
**Popov**VS. Voskresenskii DN. **Title**- Distribution of a vacuum charge near super-charged nuclei.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.72, no.3, March 1977, pp.834-50. USSR.
**Abstract**- The distribution of a vacuum charge near super-critical nuclei (Ze/sup 2/[right angle bracket][right angle bracket]1) is found. The calculation is performed in the Thomas-Fermi approximation extended to the relativistic case. For Ze/sup 3/[right angle bracket]or approximately=1 the electron shell penetrates the nucleus and almost completely screens its charge. For Ze/sup 3/[right angle bracket][right angle bracket]1 the supercharged nucleus is an electrically neutral plasma with equal concentrations of electrons, protons and neutrons (when N=Z). The potential within the nucleus is constant and equals -V/sub 0/=-(3 pi /sup 2/n/sub p/)/sup 1/3/ approximately=-1.94 m/sub pi /c/sup 2/. Near the boundary of the nucleus there is a transition region with a width which is independent of Z. The electric field and surface charge are located in this region. The Coulomb energy of the nucleus, which impedes its stability, is reduced as a result of screening and the E/sub Q/ varies as Z/sup 5/3/ dependence is replaced by a Z/sup 2/3/ dependence. The Thomas-Fermi equation for a neutral atom is also considered.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Engng. Phys. Inst., Moscow, USSR.
**Title**- Bound states near the limit of the lower continuum (fermion case).
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.27, no.2, May 1976, pp.204-16. USSR.
**Abstract**- The effective range approximation is generalized to the case of the Dirac equation. Formulas are obtained that express the parameters in the expansion of the S-matrix as epsilon to -mc/sup 2/ in terms of the wave function of the level at the critical point. These formulas are used to investigate the entry of levels of the discrete spectrum into the lower continuum. The critical charge of the nucleus for muons is calculated (the case R[right angle bracket][right angle bracket]h(cross)/mc).

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Engng. Phys. Inst., Moscow, USSR.
**Title**- Bound states near the limit of the lower continuum (boson case).
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.27, no.1, April 1976, pp.81-93. USSR.
**Abstract**- The well-known effective range approximation in nonrelativistic scattering theory is generalized to the case of scalar particles satisfying the Klein-Gordon equation. Exact expressions are obtained for the parameters of the expansion of the S-matrix and the energy of levels near the limit of the lower continuum in terms of the wave function at the point at which a bound state arises for antiparticles (i.e., at epsilon =-mc/sup 2/). These expressions are used to investigate the motion of the levels near the limit epsilon =-mc/sup 2/ for different values of the angular momentum l. It is shown that the p-level curve can 'bend' for potentials with sharp edge; this was known previously only for s-levels. A number of exactly solvable examples is considered. In particular, the exact solution of the Klein-Gordon equation for the s levels in the Hulthen potential is investigated in detail, together with the passage to the limit of unscreened Coulomb potential.

**Author**- Eletskii VL.
**Popov**VS. **Title**- Critical nuclear charge for higher levels. (Numerical calculations and the quasiclassical approach).
**Source**- Yadernaya Fizika, vol.25, no.5, 1977, pp.1107-12. USSR.
**Abstract**- Numerical results are presented for the critical nuclear charges corresponding to all the electronic levels crossing the boundary of the lower continuum at Z alpha [left angle bracket]9. Quasi-classical formulae for the number of states in the lower continuum are obtained and compared with the numerical calculation. Integral representations are obtained for the Macdonald function with an imaginary index and its derivative, that are appropriate for numerical computations at any values of the argument and the index.

**Author**- Lisin VI. Marinov MS.
**Popov**VS. **Institution**- Inst. for Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Critical distance for the electron two-center problem.
**Source**- Physics Letters B, vol.69B, no.2, 1 Aug. 1977, pp.141-2. Netherlands.
**Abstract**- Critical distance R/sub cr/ for the spontaneous positron creation in heavy-ion collisions is calculated precisely for point-like nuclei with 85[left angle bracket]or=Z[left angle bracket]or=126. For example, in case of U-U collision R/sub cr/=38.4 fm. Rapid convergence of the method is clearly demonstrated.

**Author****Popov**VS. Eletskii VL. Turbiner AV.**Title**- Summing asymptotic series in quantum field theory.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.26, no.3, 5 Aug. 1977, pp.193-7. USSR.
**Abstract**- Examines a method of summing series in the perturbation theory with factorially increasing coefficients a/sub k/. A connection is established between the asymptote of a/sub k/ when k to infinity and the nature of the special characteristics of the sum of the series. The boundaries of the range of applicability of the improved perturbation theory are found. The results obtained are tested on a number of physics problems for which exact solutions are known.

**Author****Popov**VS.**Title**- New expressions for the invariant operators of the unitary groups.
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.29, no.3, Dec. 1976, pp.357-69. USSR.
**Abstract**- The invariant operators or (Casimir operators) for the unitary groups U(n) and SU(n) are considered. The eigenvalues of these operators for arbitrary irreducible representation are expanded with respect to standard power sums S/sub k/. Expansion are obtained for the coefficients ( beta )/sub nu /( nu ); they hold for arbitrary rank n-1 of the group and arbitrary order p of the invariant operator. These expressions considerably simplify the calculation of the eigenvalues of the invariant operators (especially for large p), which is demonstrated by a number of examples.

**Author****Popov**VS. Eletskii VL. Marinov MS.**Title**- Critical charge for anomalous nuclei.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.73, no.4, Oct. 1977, pp.1241-57. USSR.
**Abstract**- The dependence of the critical charge of nuclei on the proton density is found. The values of Z/sub cr/ for anomalous nuclei are obtained. The calculations of Z/sub CR/ are carried out by taking into account the diffusion of the nuclear boundary and also the effect of screening of the Coulomb nuclear potential by the electron shell. The Thomas-Fermi statistical method is employed for describing the electron density in the shell. Two cases are considered: 1) screening by an ordinary electron shell formed by electrons occupying levels of the discrete spectrum (-m[left angle bracket] epsilon [left angle bracket]m); 2) screening by a vacuum shell formed by electrons located at levels with energies of the atom, q=(Z-N)/Z is also found. The properties of electron states at the critical point are considered in detail. Asymptotic formulas for solution of the Thomas-Fermi equation are derived for two extreme cases: q to O (weakly ionized atom) and q to 1.

**Author****Popov**VS. Eletsky VL. Turbiner AV.**Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Borel summation of perturbation series in quantum mechanics and field theory.
**Source**- Physics Letters B, vol.72B, no.1, 5 Dec. 1977, pp.99-102. Netherlands.
**Abstract**- The Borel summation of factorial divergent perturbation series is considered. The relation between the asymptotics of the coefficients in the series and the character of the exact function singularity is established. The applicability limits for the improved perturbation theory are obtained. The results are tested on a number of physical problems for which the exact solutions are available.

**Author**- Eletsky VL. Voskresensky DN.
**Popov**VS. **Title**- Quasiclassical approximation and the Thomas-Fermi method for Z>137.
**Source**- Yadernaya Fizika, vol.26, no.5, 1977, pp.994-1004. USSR.
**Abstract**- The quasiclassical approximation is applied to solve the problem of the electron in the Coulomb field of the nucleus with the charge Z when Ze/sup 2/[right angle bracket]1 and to describe the multielectronic shell of the supercritical (Ze/sup 2/[right angle bracket][right angle bracket]1) atom. An approximate method based on the condition R[left angle bracket][left angle bracket]r (the short nuclear range approximation) is developed to calculate the physical characteristics of the electron states at the energy close to the boundary of the lower continuum. The accuracy of these methods is discussed. The 'fall to the centre' in the Thomas-Fermi relativistic equation for the electronic shell of the supercritical atom is studied.

**Author**- Eletskii VL.
**Popov**VS. **Title**- The Thomas-Fermi method for Z>137.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.73, no.6, Dec. 1977, pp.2046-59. USSR.
**Abstract**- A solution is obtained for the relativistic Thomas-Fermi equation for the electron cloud density near supercritical (Ze/sup 2/[right angle bracket][right angle bracket]1) nuclei. The cases of a vacuum ion (supercritical nucleus surrounded by a vacuum shell), neutral atom and positive ion are considered. The angular momentum distribution of the vacuum shell electrons is found. For Ze/sup 3/[left angle bracket][left angle bracket]1 (weak screening) the electron shell of the supercritical atom consists of two parts which almost do not overlap. These are the vacuum shell with a radius r/sub a/ approximately Ze/sup 2//2 and the outer shells with a radius r/sub e/ approximately (Z/sup 1/3/e/sup 2/)/sup -1/. Here r/sub a//r/sub e/ approximately (Ze/sup 3/)/sup 4/3/[left angle bracket][left angle bracket]1. 'Falling to the center' is investigated in the relativistic Thomas-Fermi equation. The nature of the singularities of the equation solutions at r to 0 is determined by the renormalization group technique. A differential equation for the Gell-Mann-Low function beta ( mu ) is obtained and its asymptotic values for mu to 0 and mu to infinity are found. A numerical solution is obtained for all values of the effective coupling constant mu .

**Author****Popov**VS. Yeletskii VL. Turbiner AV.**Title**- Higher orders in perturbation theory and the summation of series in quantum mechanics and field theory.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.74, no.2, Feb. 1978, pp.445-65. USSR.
**Abstract**- The Borel method for summation of the perturbation theory series with factorially increasing coefficients is considered. The relation between the asymptotic value of the coefficients of the series a/sub k/, for k to infinity , and the singularity of the sum is established. An improved perturbation theory is developed and the limits of its applicability are found. The results are verified for a number of physical problems for which the coefficients of the perturbation theory series exhibit factorial growth and the exact solutions are known. By applying the improved perturbation theory to the scalar field theory g phi /sup 4//4] it is possible to determine the behavior of the Gell-Mann-Low function psi (g) for 0[left angle bracket]g/16 pi /sup 2/[left angle bracket]or approximately=0.3. In this interval psi (g) is a monotonously growing function of g.

**Author**- Mur VD.
**Popov**VS. Voskresenskii DN. **Title**- On levels in the lower continuum.
**Source**- Yadernaya Fizika, vol.27, no.2, 1978, pp.529-41. USSR.
**Abstract**- Quasistationary levels belonging to the lower (negative) continuum of solutions of the Dirac equation are considered. Application of the WKB approximation enables one to solve the problem of the real part of the level energy completely. Explicit formulae are obtained for the level spectrum with the energy epsilon [left angle bracket]-mc/sup 2/, their numbers and the angular momentum distribution are found. Comparison with the numerical calculation shows that the WKB method has a reasonable accuracy even at zeta =Ze/sup 2//h(cross)c[right angle bracket]or approximately=2. The change of the asymptotics (r to infinity ) of the solutions of the Dirac equation at crossing the boundary of the lower continuum is explicated.

**Author**- Mur VD.
**Popov**VS. Voskresenskii DN. **Title**- The WKB method for Z>137.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.28, no.3, 5 Aug. 1978, pp.140-4. USSR.
**Abstract**- WKB approximation formulae are obtained for Dirac's equations in a strong ([right angle bracket]2 M/sub e/C/sup 2/ electron binding energy) external field, and used to calculate the energies and widths of the quasistationary states in the lower continuum. A preexponential factor for the probability of spontaneous positron production is also found.

**Author**- Eletskii VL.
**Popov**VS. **Title**- Calculation of the Gell-Mann-Loy function in scalar field theory.
**Source**- Yadernaya Fizika, vol.28, no.1, 1978, pp.268-70. USSR.
**Abstract**- Lipatov's work (1976, 1977) suggests a quasi-classical method of calculating functional integrals in perturbation theory, and analyses the coefficients of the Gell-Mann-Loy function in scalar field theory. It is proposed that Lipatov's method of calculating the asymptotic coefficients should be used in conjunction with the Pade-Borel approximation for accurate and economical computation of the Gell-Mann-Loy function. An example is presented where the function is calculated for a scalar field g phi /sup 4//4].

**Author**- Eletsky VL.
**Popov**VS. **Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- On the Pade-Borel summation of perturbation series.
**Source**- Physics Letters B, vol.77B, no.4-5, 28 Aug. 1978, pp.411-14. Netherlands.
**Abstract**- Pade-Borel summation of perturbation series with factorially growing coefficients is considered. Application of this method to scalar field theory g phi /sup 4//4] allows one to compute the Gell-Mann-Low function psi (g) at 0[left angle bracket]g/16 pi /sup 2/[left angle bracket]0.6. In this interval it grows monotonously with the coupling constant g and has no zeros.

**Author**- Mur VD.
**Popov**VS. **Title**- Semiclassical approximation for the Dirac equation in strong fields.
**Source**- Yadernaya Fizika, vol.28, no.3, 1978, pp.837-49. USSR.
**Abstract**- A consistent scheme of the semiclassical approximation for the Dirac equation in strong external field is developed. The approximate wave function is found both in classically allowed region and in the forbidden region and the joining conditions in the turning point are obtained. Equations are obtained for the energies of quasistationary levels lying in the lower continuum of solutions of the Dirac equation. The method is applied to relativistic corrections to the Bohr-Sommerfeld quantization rule, to calculation of the preexponential factor in probability of the position spontaneous production by the Coulomb field of a nucleus with charge Z[right angle bracket]Z/sub cr/. Semiclassical formulae are obtained for the number of levels dropped into the lower continuum.

**Author**- Eletskii VL.
**Popov**VS. **Title**- Summing of perturbation theory series by Pade and Pade-Borel methods.
**Source**- Yadernaya Fizika, vol.28, no.4, 1978, pp.1109-22. USSR.
**Abstract**- The series in the perturbation theory and quantum mechanics often have coefficients which increase as factorials a/sub k/ approximately const.(k alpha )]a/sup k/k/sup beta /, k to infinity . Therefore the convergence radius of these series is zero. The Pade and Pade-Borel methods allow to find the sum of the perturbation theory series psi (g)= Sigma a/sub k/(-g)/sup k/ (in a certain g interval) using the first coefficients a/sub k/ and their asymptotic behaviour at k to infinity . These methods (especially the Pade-Borel method) have essential advantages as compared to the ordinary and improved perturbation theory. The results are illustrated by a number of physical problems for which the coefficients a/sub k/ increase as factorials and at the same time the exact solutions are known. Application of the Pade-Borel method to the scalar field theory with the interaction H/sub int/=(g/4]) integral phi /sup 4/d/sup 4/x allows to find the Gell-Mann-Low function psi (g) at 0[left angle bracket]g/16 pi /sup 2/[left angle bracket]0.6. In this range psi (g) rises monotonously with increase of the coupling constant g and has no zeroes.

**Author****Popov**VS.**Institution**- Inst. of Theoretical & Experimental Phys., USSR.
**Title**- Superheavy atoms.
**Source**- Izvestiya Akademii Nauk SSSR Seriya Fizicheskaya, vol.41, no.12, 1977, pp.2577-84. USSR.
**Conference Information**- Proceedings of the All-Union Conference 'Modern Theory of Atoms and Atomic Spectra. Leningrad, USSR. 1-4 Feb. 1977.
**Abstract**- A review is given of the present state of the theory of superheavy (atomic number greater than 137) atoms. Problems reviewed include the critical charge of the nucleus, entry of discrete levels into the lower (negative) continuum, properties of the electron shells of supercritical atoms and spontaneous creation of positrons in slow collisions between heavy ions.

**Author**- Dolgov AD.
**Popov**VS. **Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Modified perturbation theories for an anharmonic oscillator.
**Source**- Physics Letters B, vol.79B, no.4-5, 4 Dec. 1978, pp.403-5. Netherlands.
**Abstract**- A simple method of higher-order calculations in perturbation theory (PT) for an arbitrary gv(r) anharmonic oscillator is presented. The method is based on using the Riccati equation instead of the Schrodinger one. The structure of the perturbation series for the energy eigenvalues is investigated. If v(r) to exp(br/sup v/) as r to infinity , then the perturbation series is not Borel summable for 0[left angle bracket]v[left angle bracket]2. If v[right angle bracket]or=2 the PT expansion in powers of g does not exist. A new convergent version of PT valid for any g, 0[left angle bracket]g[left angle bracket] infinity , is proposed.

**Author****Popov**VS. Eletsky VL. Mur VD. Voskresensky DN.**Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- WKB approximation for the Dirac equation at Z>137.
**Source**- Physics Letters B, vol.80B, no.1-2, 18 Dec. 1978, pp.68-72. Netherlands.
**Abstract**- The WKB formulae are obtained for the Dirac equation in a strong external field, i.e. a field in which the electron binding energy exceeds 2m/sub e/c/sup 2/ and the energy levels cross the lower continuum boundary. The energy and the width of the quasistationary states in the lower continuum are calculated as well as the preexponential factor in the probability of spontaneous positron production by the Coulomb field. A simple quasiclassical expression for the number of levels immersed into the lower continuum is obtained. Other applications of the WKB method to the theory of supercritical atoms are briefly discussed.

**Author****Popov**VS.**Institution**- Inst. of Theoretical & Experimental Phys., USSR.
**Title**- Invariant operators for the classical groups.
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.32, no.3, Sept. 1977, pp.344-7. USSR.
**Abstract**- The eigenvalues of invariant operators of arbitrary order for all classical groups are calculated. The eigenvalues are represented as polynomials of the power sums S/sub k/.

**Author**- Dolgov AD.
**Popov**VS. **Title**- Higher orders and the structure of perturbation theory for an anharmonic oscillator.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.75, no.6, Dec. 1978, pp.2010-26. USSR.
**Abstract**- A simple method is developed for the calculation of the higher orders in perturbation theory (PT) in powers of g for a D-dimensional isotropic oscillator with arbitrary anharmonicity gv(r). The method is based on a transition from the Schrodinger equation to the Riccati equation. In the important particular case of power law anharmonicity when v(r)=r/sup 2N/, N=2, 3, 4, ..., all the xi /sub n/(r) terms of the PT series become polynomials and hence calculation of the higher orders of perturbation theory is considerably simplified. A new variant of PT is proposed which converges for all values of the coupling constant g, 0[left angle bracket]or=g[left angle bracket] infinity . The structure of the PT series for the energy levels is investigated for potentials with power (v(r) approximately r/sup v/) and exponential (v(r) approximately exp(ar/sup 2//sup v/)) growth at infinity.

**Author**- Kudryavtsev AE.
**Popov**VS. **Title**- On the theory of the nuclear shift of proton-antiprotonic atom levels.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.29, no.5, 5 March 1979, pp.311-16. USSR.
**Abstract**- A formula is obtained, expressing the shift in coulomb levels of hadronic atoms in terms of nuclear potential scattering length (without a small shift presumption). Experimental data on the pp-atom 1s-level shift point to a possible bound state with a binding energy of about 0.9 MeV and width [left angle bracket]200 keV in the pp system.

**Author****Popov**VS. Voskresenskii DN. Eletskii VL. Mur VD.**Title**- The WKB method for Z>137 and its application to the theory of supercritical atoms.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.76, no.2, Feb. 1979, pp.431-59. USSR.
**Abstract**- Formulas for the Dirac equation in a strong external field are obtained in the WKB approximation. A field is considered strong if the electron coupling energy exceeds 2m/sub e/c/sup 2/ and the discrete spectrum levels may be lowered into the continuum. The wave functions in the classically allowed and forbidden regions are found and the conditions for matching them on transition through the turning point are obtained. The WKB method is applied to the following problems: 1) generalization of the Bohr-Sommerfeld quantization conditions with allowance for relativistic effects and the spin; 2) calculation of the pre-exponential factor in the probability for spontaneous positron production; 3) the energy and width of the levels in the relativistic two-center problem; 4) effect of screening on the critical distance R/sub cr/ between the colliding nuclei; 5) tunneling through a nonspherical barrier and the positron angular distribution; 6) allowance for the finite velocity of nuclei. Some other problems which lie outside the limits of applicability of the WKB method are also discussed e.g. the electron spectrum Ze/sup 2/=1 and the deep levels in the lower continuum (Ze/sup 2/[right angle bracket][right angle bracket]1, j[left angle bracket]or approximately=(Ze/sup 2/)/sup 1/2/).

**Author**- Dolgov AD.
**Popov**VS. **Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- The anharmonic oscillator and its dependence on space dimensions.
**Source**- Physics Letters B, vol.86B, no.2, 24 Sept. 1979, pp.185-8. Netherlands.
**Abstract**- The perturbation series (PS) in powers of g for the D-dimensional gr/sup 4/ anharmonic oscillators is considered. High-order terms of the ground-state-energy E(g,D) expansion are explicitly calculated with the help of recurrence relations among integers. The rate of approach of the PS coefficients to the asymptotic value as a function of D is investigated. For the case of D=0 explicit solutions are obtained and analytic properties of E(g,0) are briefly discussed. For D[right angle bracket][right angle bracket]1 the first terms in the expansion of E(g,D) in powers of D/sup -1/ are found.

**Author**- Alliluev SP. Eletsky WL.
**Popov**VS. **Institution**- Moscow Phys. Tech. Inst., Moscow, USSR.
**Title**- The high orders of perturbation theory for the Stark effect in hydrogen.
**Source**- Physics Letters A, vol.73A, no.2, 3 Sept. 1979, pp.103-6. Netherlands.
**Abstract**- An effective method is developed to calculate the high orders of the perturbation series (PS) for the Stark shift of the energy levels in the hydrogen atom. The method is based on the use of the Riccati equation instead of the Schrodinger equation. The PS coefficients alpha /sub k/ (hyperpolarizabilities) are given by simple recurrence relations among integers. For the ground state alpha /sub k/ are calculated exactly for k[left angle bracket]or=14 and numerically up to k=160. The asymptotic behaviour of alpha /sub k/ when k to infinity is discussed for the energy eigenvalues E/sub n/, 1[left angle bracket]or=n[left angle bracket]or=5. With the results obtained the energy and the width of the ground state are determined for external fields E[left angle bracket]or approximately=0.1 au.

**Author****Popov**VS. Kudryavtsev AE. Mur VD.**Title**- Nuclear level shift and radiative transitions in a proton-antiproton atom.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.77, no.5, 1979, pp.1727-50. USSR.
**Abstract**- A formula for the Coulomb level shifts in a proton-antiproton atom is derived in terms of the scattering length for a strong potential V/sub s/. The rearrangement of the atomic spectrum when a bound state in the V/sub s/ potential appears is discussed on the basis of the formula. The relation between the level shift and effective radius expansion is pointed out. The mean radius of the s-state for an arbitrary value of the shift and the probabilities for radiative E1 transitions between p- and s-levels are calculated. From the experimental data on the 1s-level shift in the pp-atom it is concluded that a bound Qs-state (quasinuclear meson) with a binding energy epsilon approximately=1 MeV and width Gamma [left angle bracket]or approximately=200 keV may exist. Calculations show that the probabilities for 2p to 1s and 2p to Qs radiative transitions are approximately equal. This indicates that it should be feasible to observe experimentally the Qs-level by investigating the gamma -ray spectrum produced in transitions between pp atomic levels.

**Author**- Mur VD.
**Popov**VS. **Title**- The theory of a nuclear shift of levels in heavy atoms.
**Source**- Yadernaya Fizika, vol.31, no.3, 1980, pp.617-25. USSR.
**Abstract**- The results of Kudryavtsev et al. (1979) are generalized to the case of the relativistic Coulomb problem when the parameter zeta =Za approximately=Z/137 is not supposed to be small. The method of matching derivatives is used, the Klein-Gordon and Dirac equations being solved exactly outside the nucleus (r[right angle bracket]r/sub N/). For mr/sub N/[left angle bracket][left angle bracket]1 the nuclear shift of the levels is written in an explicit analytical form. The derived formulae are applied to calculate the critical distance in heavy ion collisions (taking into account finite size of the colliding nuclei).

**Author**- Lisin VI. Marinov MS.
**Popov**VS. **Institution**- Inst. for Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Critical electron state in heavy-ion collisions.
**Source**- Physics Letters B, vol.91B, no.1, 24 March 1980, pp.20-2. Netherlands.
**Abstract**- The results of a calculations of the electron wave function for the critical state epsilon =-m/sub e/c/sup 2/) and of some related quantities are presented. The effect of the finite nuclear size on the critical internuclear distance R/sub cr/ is taken into account. The calculations are performed by means of the variational method for nuclei with charges 85[left angle bracket]or=Z[left angle bracket]or=100. The error of the WKB method for strong Coulomb fields is found to be a few percent.

**Author**- Mur VD. Kudryavtsev AE.
**Popov**VS. **Title**- On the influence of the nn-channel on the properties of a proton-antiproton atom.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.31, no.7, 5 April 1980, pp.429-33. USSR.
**Abstract**- The influence of the second channel (nn) on nuclear level shifts and the probabilities of radiation transitions in a pp atom are studied. The obtained formulae can also be used on other hadronic atoms ( Sigma /sup -/p, K/sup -/p etc.)

**Author**- Eletskii VL. Mur VD.
**Popov**VS. **Title**- On the WKB method for Z>137.
**Source**- Doklady Akademii Nauk SSSR, vol.249, no.1-3, Nov. 1979, pp.329-33. USSR.
**Abstract**- The authors study the WKB method for a strong external field (in the Coulomb problem this corresponds to the region Z[right angle bracket]137). For the relativistic wavefunctions they obtain approximations that can be used in the neighborhood of a turning point and also consider the question of the accuracy of the WKB method.

**Author**- Alliluev SP. Eletsky VL.
**Popov**VS. Weinberg VM. **Institution**- Moscow Phys. Tech. Inst., Moscow, USSR.
**Title**- Stark effect in hydrogen: high orders of perturbation theory for excited states.
**Source**- Physics Letters A, vol.78A, no.1, 7 July 1980, pp.43-6. Netherlands.
**Abstract**- The method proposed previously by the authors (1978-79) for computation of high order terms of perturbation series (PS) is generalized to excited states with wave-functions having nodes. The computation of the PS coefficients E/sub k/ for the Stark shift in hydrogen is reduced to solving a system of recurrence relations. With these relations exact values of E/sub k/ are calculated up to k=12 for all states with the principal quantum number n=2 as well as for the states with n/sub 1/=n/sub 2/=0, mod m mod =n-1 and n[left angle bracket]or=5. Numerical values of E/sub k/ are obtained for k[left angle bracket]or=100. This makes it possible to investigate the approach of E/sub k/ to their asymptotics E/sub k/ and to determine power corrections to E/sub k/.

**Author**- Eletsky VL.
**Popov**VS. **Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- On perturbation theory for the gr/sup 2N/ anharmonic oscillator.
**Source**- Physics Letters B, vol.94B, no.1, July 1980, pp.65-8. Netherlands.
**Abstract**- Perturbation series (PS) in powers of the coupling constant g for D-dimensional anharmonic oscillator with power anharmonicity gr/sup 2N/ is considered. The high order PS coefficients epsilon /sub k/ for the ground state energy are calculated explicitly with the help of recurrence relations among integers. The rate of approach of epsilon /sub k/ to their asymptotics epsilon /sub k/ as a function of space dimension D is discussed.

**Author**- Eletskii VL.
**Popov**VS. **Title**- The divergence of the perturbation theory series for the Stark effect and summation of the series.
**Source**- Doklady Akademii Nauk SSSR, vol.250, no.1-3, Jan. 1980, pp.74-7. USSR.
**Abstract**- Effective methods for calculating the higher orders of perturbation theory have been found in recent years in quantum mechanics and field theory. A number of problems of quantum mechanics are known for which the perturbation-theory coefficients possess a factorial increase and at the same time one can obtain (analytically or numerically) an exact solution of the Schrodinger equation. The anharmonic oscillator and the Stark effect are classical examples. The results for the Stark effect (a hydrogen atom in a uniform electric field) are described.

**Author**- Dolgov AD. Eletskii VL.
**Popov**VS. **Title**- A new approach to perturbation theory for a discrete spectrum (anharmonic oscillator).
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.79, no.5, Nov. 1980, pp.1704-18. USSR.
**Abstract**- An effective method is developed for calculating the higher orders of perturbation theory (PT) for discrete spectrum. The method is based on the transition from the Schrodinger equation to the Riccati equation for the logarithmic derivative of the wave function. For the polynomial potentials (and in particular for the case of an anharmonic oscillator with a gr/sup 2N/ nonlinearity, N=2, 3, 4,...) the calculation of higher orders of PT is reduced to simple recurrence relations. Using these relations the coefficients in the PT series for the energy levels are found exactly for k[left angle bracket]or=10 and numerically up to k approximately 200 (k is the PT order). A rapidly converging PT with the behavior of the potential at singular points taken into account is developed on the basis of the Riccati equation.

**Author****Popov**VS. Kudryavtsev AE. Lisin VI. Mur VD.**Title**- On the theory of nuclear level shifts in hadronic atoms.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.80, no.4, 1981, pp.1271-87. USSR.
**Abstract**- Due to strong short range interaction the Coulomb levels of hadronic atoms experience shifts and broadening. The presence in the problem of a small parameter r/sub e//a/sub B/ makes it possible to develop an analytic theory of the phenomenon (r/sub e/ is an effective range of the strong potential and a/sub B/=h(cross)/sup 2//m/sup e2/ is the Bohr radius). Terms of the order of r/sub e/, the proton form factor and also the effect of the second channel (which is close in energy) are taken into account in the equation for the level shifts. The shifts and broadening of the s-levels and the restructuring of the atomic spectrum are investigated in the presence of absorption, i.e. for a complex scattering length. The region of applicability of perturbation theory in terms of the scattering amplitude is determined. The theory is extended to states with nonvanishing angular momenta.

**Author**- Eletsky VL.
**Popov**VS. Weinberg VM. **Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Logarithmic perturbation theory for the screened Coulomb potential.
**Source**- Physics Letters A, vol.84A, no.5, 3 Aug. 1981, pp.235-8. Netherlands.
**Abstract**- Logarithmic perturbation theory (PT) is formulated for the screened Coulomb potential in powers of the screening parameter mu . Recurrence relations are obtained which allow one to compute successively the PT series coefficients E/sub k/ for the ground state energy E=E( mu ). On a number of examples the authors demonstrate that Pade summation of the PT series makes it possible to obtain the energy in the strong coupling region.

**Author**- Alliluev SP. Vainberg VM. Eletskii VL.
**Popov**VS. **Title**- Logarithmic perturbation theory for excited levels.
**Source**- Doklady Akademii Nauk SSSR, vol.254, no.4-6, Oct. 1980, pp.1392-6. USSR.
**Abstract**- A significant simplification of higher-order perturbation-theory (PT) calculations in quantum mechanics has recently been achieved. For perturbations of the form V(x)= Sigma /sub -2[left angle bracket]or=k[left angle bracket]or=N/c/sub k/x/sup k/, which are frequently encountered in practice, the calculation of the coefficients in the PT series for the energy levels is reduced to recurrence relations that are convenient for numerical calculation. In this way about 150 orders of PT were found for the anharmonic gr/sup 2N/-oscillator and the Stark effect in hydrogen. The authors continue the study of this method and generalize it to the case of excited states whose wave functions have nodes.

**Author**- Kudryavtsev AE. Lisin VI.
**Popov**VS. **Title**- Coulomb systems with short-action distance: Sigma /sup -/p atom and the resonance states of light nuclei.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.34, no.5, 5 Sept. 1981, pp.292-6. USSR.
**Abstract**- The equation connecting the shift and breadth of atomic levels with strong scattering length, is generalised for the case of complex pulses k and arbitrary values of momentum l and zeta =-Z/sub 1/Z/sub 2/. Estimates are presented for the shift and breadth of the s-levels for Sigma /sup -//sub p/ atom. The connection between the position and width of resonances in the case of Coulombic scattering is investigated.

**Author**- Veinberg VM. Eletskii VL.
**Popov**VS. **Title**- Logarithmic perturbation theory for screened Coulomb potential and charmonium potential.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.81, no.5, 1981, pp.1567-82. USSR.
**Abstract**- Logarithmic perturbation theory (PT) for screened Coulomb potential is considered. Calculation of higher PT orders of the level energy E/sub k/ is reduced to recurrent relations which are valid for any type of screening and are very convenient for computer calculations. They are employed to calculate the PT coefficient of E/sub k/ up to k=100 for a number of physically interesting cases such as the Yukawa potential, quarkonium potential etc. The approach of E/sub k/ at k to infinity to the asymptotics. E/sub k/, defined by the quasiclassical approximation, is discussed. The problem of summation of the divergent PT series is considered. By employing the Pade approximants and the asymptotics E/sub k/ it is possible to determine with a high accuracy the energy level E(g) for the potentials mentioned above in a range which considerably exceeds the range of applicability of PT, including the strong coupling region.

**Author****Popov**VS.**Title**- Quantum electrodynamics of superforce fields.
**Source**- Priroda, no.10, Oct. 1981, pp.14-22. USSR.
**Abstract**- The author firstly describes the progress and results of investigations of the phenomena of quantum electrodynamics, i.e. the electromagnetic interactions of elementary particles, since Dirac's formulation in 1928 of his relativistic wave equation. Physical effects in superheavy nuclei are then described, and it is shown how these effects may be analysed in greater detail by applying the so-called 'method of effective potential', in which the Dirac equation is replaced by the simpler one of Schroedinger. Finally an account is given of the mechanism of formation of positrons (positively charged electrons) by the high-speed interaction of heavy nuclei, as in the Darmstadt experiment, the results of which confirm the applicability of the Dirac equation for describing the movement of particles in very powerful electromagnetic fields.

**Author**- Alliluev SP. Veinberg VM. Eletskii VL.
**Popov**VS. **Title**- Logarithmic perturbation theory and its application to the Stark effect in the hydrogen atom.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.82, no.1, 1982, pp.77-90. USSR.
**Abstract**- The technique considered by S.P. Alliluev et al. (see Phys. Lett. A, vol.78, p.43, 1980) for calculating of higher orders in perturbation theory (PT) in quantum mechanics is extended to excited states whose wave functions have nodes. Calculation of the E/sub k/ coefficients of the PT series for the level energies reduces to recurrent relations which are convenient for numerical computation. As an example the Stark effect in the hydrogen atom is considered for states with n=1 or 2 and also for all states with n/sub 1/=n/sub 2/=0. The asymptotic value for E/sub k/ for k to infinity which depends on the level width in weak fields is obtained. Summation of the divergent PT series makes it possible to determine the Stark shift and the level width in a strong electric field. The asymptotic value of the energy E(E) in extremely strong fields is found for a model which can be solved exactly (Stark effect for a one-dimensional delta -potential).

**Author****Popov**VS. Weinberg VM.**Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- On summation of perturbation series in quantum mechanics.
**Source**- Physics Letters A, vol.90A, no.3, 28 June 1982, pp.107-9. Netherlands.
**Abstract**- Summation of a divergent perturbation series for a funnel-like potential is considered. It is done by the Pade-Borel method with the transformation of the variable which moves the nearest singularity of the Borel transform to infinity. The method makes it possible to compute the energy E(g) with a good accuracy for all physical values of the coupling constant g, including the behaviour of E(g) as g to infinity .

**Author**- Vainberg VM.
**Popov**VS. **Title**- The summation of perturbation series.
**Source**- Doklady Akademii Nauk SSSR, vol.264, no.1-3, May 1982, pp.330-3. USSR.
**Abstract**- The authors consider the summation of the perturbation series for the potential V(r)=-r/sup -1/+gr, h(cross)=m=e=1, They show that, if one knows exactly the first few coefficients of the perturbation series and their asymptotic form E/sub k/ for k to infinity , one can determine the energy E(g) not only for g to 0 but also in the strong coupling regime.

**Author****Popov**VS. Kudryavtsev AE. Lisin VI. Mur VD.**Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Relation between scattering lengths and effective radii in the charged and neutral channels.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.36, no.6, 20 Sept. 1982, pp.207-10. USSR.
**Abstract**- The Coulomb correction to the strong scattering length a/sub l//sup (s)/ and the effective radius r/sub l//sup (s)/ are calculated for states with arbitrary angular momentum l. This allows relating the position and width near threshold resonances in charged and neutral channels for the lightest nuclei (/sup 5/He and /sup 5/Li, etc.). The Coulomb correction to a/sub l//sup (s)/ is especially large for the s wave, while the correction to r/sub l//sup (s)/ is large for the p wave, when they contain the large logarithm ln(a/sub B//r/sub 0/), where a/sub B/ is the Bohr radius and r/sub 0/ is the range of nuclear forces.

**Author**- Alliluev SP. Vainberg VM.
**Popov**VS. **Title**- Higher orders of perturbation theory for the screened Coulomb potential.
**Source**- Doklady Akademii Nauk SSSR, vol.265, no.1-3, July 1982, pp.597-601. USSR.
**Abstract**- The calculation of the higher orders of perturbation theory in quantum mechanics has been significantly advanced recently by the development of logarithmic perturbation theory. The authors consider its application to the screen Coulomb potential.

**Author**- Mur VD. Kudryavtsev AE.
**Popov**VS. **Institution**- Inst. of Theoretical & Experimental Phys., State Comm. on Use of Atomic Energy, USSR.
**Title**- Coulomb corrections to scattering length and effective range.
**Source**- Yadernaya Fizika, vol.37, no.6, June 1983, pp.1417-28. USSR.
**Abstract**- The authors consider the problem of determination of the purely nuclear scattering length a/sub s/ (pertaining to a strong-interaction potential V/sub s/ with the Coulomb interaction switched off) from the nuclear-Coulomb scattering length a/sub cs/ that is directly measured in experiment. The difference between a/sub s/ and a/sub cs/ is especially large if the potential V/sub s/ has a level (real or virtual) with energy close to zero. For such a case they have obtained the formulae relating the scattering lengths a/sub s/ and a/sub cs/ and the effective ranges r/sub s/ and r/sub ss/. The results are generalized to states with arbitrary angular momentum l. They show that the Coulomb correction to the coefficient of k/sup 2l/ in the effective-range expansion is especially large because of the large logarithm ln(a/sub B//r/sub 0/). The Coulomb renormalization of the other terms in the effective-range expansion is of the order r/sub 0//a/sub B/, where r/sub 0/ is the range of the nuclear force and a/sub B/ is the Bohr radius. The formulae obtained are illustrated for several model potentials V/sub s/ used in nuclear physics.

**Author**- Vainberg VM.
**Popov**VS. **Title**- Summing the perturbation series for the Yukawa potential.
**Source**- Doklady Akademii Nauk SSSR, vol.272, no.1-3, Sept. 1983, pp.335-40. USSR.
**Abstract**- Over the last few years, significant progress has been made in computing higher orders of perturbation theory (PT) in quantum mechanics and field theory. Very high orders of PT for a screened Coulomb potential have been computed-in particular, for the Yukawa potential V(r)=-e/sup - mu r/r/sup -1/ (up to k=100). Since the PT series for the energy levels E/sup (nl)/( mu )= Sigma /sub k=0//sup infinity /E/sub k//sup (nl)/ mu /sup k/ has zero radius of convergence, the computation of higher orders of the PT does not by itself solve the problem of finding the energy eigenvalues E/sup (nl)/ in the strong-coupling region. The authors consider the summation of the divergent series, taking the Yukawa potential as an example.

**Author**- Kudrjavtsev AE. Mur VD.
**Popov**VS. **Institution**- Inst. of Theoretical & Experimental Phys., Moscow, USSR.
**Title**- Possible existence of K/sup -/-light nucleus bound states.
**Source**- Physics Letters B, vol.143B, no.1-3, 9 Aug. 1984, pp.41-4. Netherlands.
**Abstract**- Experimental data on the shift of the K/sup -/ /sup 4/He atom 2p-level evidence a possible existence of a loosely bound nuclear p-state in this system, its binding energy and width being epsilon approximately J approximately 0.5 MeV. The probabilities of radiative transitions to this state and its production cross section in nuclear reaction on /sup 6/Li are calculated.

**Author****Popov**VS. Weinberg VM.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR.
**Title**- On summation of perturbation series for screened Coulomb potentials.
**Source**- Physics Letters A, vol.107A, no.8, 25 Feb. 1985, pp.371-5. Netherlands.
**Abstract**- Higher orders of perturbation theory (PT) in quantum mechanics and field theory rise factorially at k to infinity (k is the PT order). The authors discuss the computation results of higher orders of PT and summation methods for divergent perturbation series (PS) for quantum mechanical problems: the Yukawa and Hulthen potentials, the funnel-shaped potential, etc. Criteria of PS summability by Pade approximants are examined.

**Author****Popov**VS. Vainberg VM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Diverging perturbation-theory series and classical mechanics.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.41, no.10, 25 May 1985, pp.439-42. USSR.
**Abstract**- A summation of the diverging perturbation-theory series yields expressions for the energies of levels (real, virtual, and quasistationary) in terms of the coefficients of the perturbation theory, E/sub k/. Taking the limit of classical mechanics reveals the asymptotic behavior of the highly excited levels with l=n-1[right angle bracket][right angle bracket]1, confirming the results of the summation of the perturbation-theory series.

**Author**- Vainberg VM.
**Popov**VS. **Title**- Summation of divergent series in perturbation theory for screened Coulomb potentials.
**Source**- Doklady Akademii Nauk SSSR, vol.279, no.1-3, Nov. 1984, pp.59-63. USSR.
**Abstract**- As a rule, the higher orders of perturbation theory (PT) in quantum mechanics and field theory increase factorially as k to infinity , and so the PT series diverge however small the coupling constant g (the Dyson phenomenon). The authors consider the summation of the PT series for the screened Coulomb potential: V(r)=-r/sup -1/f(ur). In the limit mu to 0 the energies of the levels can be expanded in formal power series: E/sup (nl)/( mu )= Sigma /sub k=0//sup infinity /E/sub k//sup (nl)/ mu /sup k/. The method of 'logarithmic perturbation theory', which has been developed in recent years, can be used effectively to calculate the PT coefficients E/sub k/. Using this method the authors have calculated E/sub k/ for various screened potentials. In a number of cases (the Yukawa, Hulthen, and funnel potentials) the terms of PT series alternate in sign, while in other cases its structure is more complicated. Because of the factorial growth of the higher orders of PT, the point mu =0 is an essential singularity for the energy E( mu ). This restricts the use of the usual polynomials of PT to the region of very small mu .

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Eng. Phys. Inst., Moscow, USSR.
**Title**- Atomic-spectrum restructuring when I not=0.
**Source**- Doklady Akademii Nauk SSSR, vol.279, no.1-3, Nov. 1984, pp.345-9. USSR.
**Abstract**- The authors discuss the restructuring of the atomic spectrum for states with l[right angle bracket]or=1. They give simple formulas that describe the shifts of l-level everywhere, including the region of intersection of terms. This analysis is, in the authors' opinion, of interest in itself, since 'atomic-spectrum restructuring' has a general quantum mechanical nature and can manifest itself not only in hadronic atoms. The phenomenon of restructuring of an atomic spectrum (for states with an arbitrary angular momentum l) is called the Zel'dovich effect.

**Author****Popov**VS.**Title**- Fast-converging perturbation theory.
**Source**- Doklady Akademii Nauk SSSR, vol.279, no.4-6, Dec. 1984, pp.1362-7. USSR.
**Abstract**- The authors analyse nonstandard perturbation theory (NPT). A detailed study of the convergence of the higher approximations of the NPT is presented. As an example, they consider the NPT for a funnel potential.

**Author****Popov**VS. Kudryavtsev AE. Lisin VI. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Shallow nuclear levels and radiative transitions in hadronic atoms.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.88, no.3, March 1985, pp.715-27. USSR.
**Abstract**- The position of a nuclear level perturbing the Coulomb spectrum can be calculated in the analytic theory of nuclear level shifts from the level shift of a hadronic atom. The is illustrated by the example of the K/sup -4/He atom. Experimental 2p-level shifts suggest that this system may have a weakly-bound p-state with binding energy and width epsilon approximately gamma approximately 0.5 MeV. The probabilities of radiative transitions to this level and the cross section for its creation in nuclear reactions on /sup 6/Li are calculated. The possible existence of states in which K/sup -/ and p are weakly bound to other light nuclei is examined. An exact solution is obtained for a model Coulomb problem with short-range interaction, and is used to determine the range of validity of the initial approximations.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Moscow Eng.-Phys. Inst., USSR.
**Title**- Zel'dovich effect for states with l=0.
**Source**- Yadernaya Fizika, vol.41, no.5, May 1985, pp.1258-71. USSR.
**Abstract**- In systems bound by Coulomb potential distorted at short distances (r approximately r/sub 0/[left angle bracket][left angle bracket]a/sub B/) by the strong interaction a rearrangement of the atomic spectrum (Zel'dovich effect) can arise. Such systems are encountered in nuclear physics (the lightest hadronic atoms: pp, Sigma /sup -/p, K/sup -4/ He, etc.), in solid state physics, etc. Qualitative features of the Zel'dovich effect are discussed for states with nonzero angular momentum l related to the existence of two regions of attraction separated by a centrifugal barrier. Simple expressions are obtained which describe atomic level shifts including the region of 'crossing terms'. The influence of the Coulomb interaction on the location of nuclear levels is considered. Expressions of the improved perturbation theory are proposed for the calculation of scattering lengths and it is shown that in the case l[right angle bracket]or=1 they are very accurate.

**Author**- Mur VD.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., State Comm. for Use of Atomic Energy, Moscow, USSR.
**Title**- Coulomb problem with short-range forces (exact solution).
**Source**- Yadernaya Fizika, vol.42, no.6, Dec. 1985, pp.1470-9. USSR.
**Abstract**- The nonrelativistic Coulomb problem with short-range interaction is considered. The model chosen for the strong potential V/sub s/(r) is the delta -function interaction on a sphere r-r/sub 0/, and this makes it possible to solve the Schrodinger equation exactly and to find explicit analytical expressions for the scattering lengths, effective ranges, phase shifts, etc. Detailed comparison of the exact solutions with approximate formulas obtained previously for an arbitrary potential V/sub s/ makes it possible to establish the limits of the range of applicability of these approximations.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., USSR.
**Title**- Coulomb problem with short-range interaction: exactly solvable model.
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.65, no.2, Nov. 1985, pp.238-49. USSR.
**Abstract**- The nonrelativistic Coulomb problem with short-range interaction is considered. The strong potential V/sub s/(r) is modeled by a delta-function interaction on a sphere r=r/sub 0/. Exact solutions to the Schrodinger equation are obtained for states with arbitrary angular momentum l together with explicit analytic expressions for the scattering lengths, effective ranges, etc. Comparison of the exact solutions with the approximate formulas established earlier for arbitrary short-range potential V/sub s/(r) makes it possible to determine the limits of applicability of these approximations.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., USSR.
**Title**- Coulomb correction to low-energy scattering.
**Source**- Doklady Akademii Nauk SSSR, vol.284, no.1-3, Sept. 1985, pp.586-91. USSR.
**Abstract**- The calculation of the Coulomb corrections to low-energy scattering parameters (scattering length a/sub l/, effective radius r/sub l/, etc.) is important in applications and has been widely discussed in the literature. The authors demonstrate that the renormalization of the coefficients in the expansion of the effective radius generally contains a 'Coulomb logarithm' lna/sub B//r/sub 0/, with the singular term being independent of the strong potential V/sub s/(r) and calculable in explicit form. This circumstance is of interest in quantum scattering theory, and is also applicable to cases where r/sub 0/[left angle bracket][left angle bracket]a/sub B/ (for example, for the lightest hadron atoms).

**Author**- Vainberg VM. Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Strong-field Stark effect.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.44, no.1, 10 July 1986, pp.9-12. USSR.
**Abstract**- The Stark shifts and the widths of the ground and excited states of the hydrogen atom are calculated. Two independent calculation methods are used: a summation of a divergent perturbation-theory series and a 1/n expansion.

**Author**- Karnakov BM. Kudryavtsev AE. Mur VD.
**Popov**VS. **Institution**- Moscow Phys. Eng. Inst., USSR.
**Title**- Zel'dovich effect in quantum mechanics.
**Source**- Nuovo Cimento della Societa Italiana di Fisica B-General Physics Relativity Astronomy & Mathematical Physics & Methods, vol.98B, ser.2, no.1, 11 March 1987, pp.63-86. Italy.
**Abstract**- In systems bound by the Coulomb potential with a short-range distortion, the reconstruction of the atomic spectrum (or Zel'dovich effect) can appear. Some peculiarities of this phenomenon for the state of nonzero angular momentum are discussed. The analytical properties of the basic equation appearing in the presence of absorption are investigated. The values of absorption at which the reconstruction of atomic levels is replaced by an oscillation regime are given. Some applications of the Zel'dovich effect for hadronic atoms and mesomolecules are considered.

**Author****Popov**VS. Vainberg VM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., State Comm. on Use of Atomic Energy, USSR.
**Title**- Higher orders of perturbation theory, classical mechanics, and the 1/n expansion.
**Source**- Yadernaya Fizika, vol.44, no.4, Oct. 1986, pp.1103-13. USSR.
**Abstract**- The higher orders of perturbation theory in quantum mechanics and field theory increase as k] for k to infinity , where k is the order. The authors discuss the results of calculations of higher orders of perturbation theory and techniques of summing divergent perturbation series for a large class of quantum mechanical problems (including the Yukawa and Hulthen potentials, the V-funnel potential, and so on). It is shown that in this way it is possible to obtain very accurate values of the energy levels (real, virtual, and quasistationary) using the perturbation series coefficients E/sub k/. By going to classical mechanics the authors find the asymptotic form of the energies of highly excited states and calculate their quantum corrections. These calculations confirm the results of summation of the perturbation series.

**Author**- Vainberg VM.
**Popov**VS. **Title**- Summation of perturbation theory series and classical mechanics.
**Source**- Doklady Akademii Nauk SSSR, vol.289, no.4-6, Aug. 1986, pp.1095-9. USSR.
**Abstract**- The problem of calculating virtual and resonance levels is constantly encountered in atomic and nuclear physics, scattering, theory, etc. The authors present an approach to this problem based on the summation of perturbation theory (PT) series. They use the results of higher-order calculations for screened Coulomb potentials.

**Author**- Mur VD.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- 1/n expansion and wave functions.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.45, no.7, 10 April 1987, pp.323-6. USSR.
**Abstract**- The 1/n expansion is used to calculate wave functions. Analytic expressions are derived. They are valid for an arbitrary smooth potential V(r), in particular, for the asymptotic coefficients in the limits r to 0 and r to infinity . Comparison with numerical calculations shows that the wave function is calculated very accurately at the origin.

**Author****Popov**VS. Mur VD. Shcheblykin AV. Weinberg VM.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR.
**Title**- 1/n-expansion for the Stark effect in strong field.
**Source**- Physics Letters A, vol.124, no.1-2, 14 Sept. 1987, pp.77-80. Netherlands.
**Abstract**- Stark shifts and widths of the ground and excited states of the hydrogen atom in a strong field E have been computed using the 1/n-expansion (up to E values comparable to the field on an electron orbit, i.e. E[right angle bracket]or approximately=n/sup -4/, and for n=1-30). Computation results for the states with parabolic quantum numbers n/sub 1/=n/sub 2/=0, m=n-1 are given and the Rydberg limit, n to infinity , is also considered.

**Author**- Vainberg VM. Orlov YuV.
**Popov**VS. Tupovtsev VV. **Title**- Virtual and resonance levels in the Yukawa potential.
**Source**- Ukrainskii Fizicheskii Zhurnal, vol.32, no.8, Aug. 1987, pp.1125-9. Ukrainian SSR, USSR.
**Abstract**- Two methods for computation of real, virtual and resonance energy eigenvalues are considered: summation of divergent perturbation series and solution of integral equations for t-matrix on physical and unphysical sheets of energy. It is shown on the example of the Yukawa potential that the energy eigenvalues given by both methods coincide with high precision. As an illustration the positions of real (deuteron) and virtual (singlet deuteron) poles of the neutron-proton scattering amplitude were calculated.

**Author****Popov**VS. Vainberg VM. Mur VD. Shcheblykin AV.**Institution**- Inst. of Theor. & Exper. Phys., Moscow, USSR.
**Title**- 1/n expansion in quantum mechanics (H atom Stark effect).
**Source**- Doklady Akademii Nauk SSSR, vol.293, no.4-6, April 1987, pp.851-5. USSR.
**Abstract**- The authors consider the 1/n expansion in quantum mechanics by including the Stark effect in the H atom. They limit the discussion to the Bohr-atom model.

**Author**- Vainberg VM. Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR.
**Title**- Stark effect for the Rydberg states of the hydrogen atom.
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.46, no.5, 10 Sept. 1987, pp.178-82. USSR.
**Abstract**- The Stark shifts and the widths of highly excited (n=15-30) states of a hydrogen atom in a strong electric field, which is higher than the classical ionization threshold of an atom, are calculated. The results of the calculations are in agreement with the experiment.

**Author**- Vainberg VM. Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- The hydrogen atom in a strong electric field.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.93, no.2, Aug. 1987, pp.450-62. USSR.
**Abstract**- A calculation is reported of Stark shifts and widths of the ground and first excited states of the hydrogen atom in a strong electric field E up to values of E comparable with the field strength on an atomic orbit. Two independent methods are used, namely, summation of divergent perturbation-theory series and the 1/n expansion. The two methods have a wide region of overlap (in the parameters n/sub 1/, n/sub 2/, m, and E). The calculations are confirmed by considering the Rydberg limit as n to infinity .

**Author**- Mur VD.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., State Comm. for Utilization of Atomic Energy, Moscow, USSR.
**Title**- The 1/n expansion and wave functions.
**Source**- Yadernaya Fizika, vol.47, no.3, March 1988, pp.697-704. USSR.
**Abstract**- The 1/n expansion is applied to the calculation of wave functions. Analytical formulas valid for an arbitrary smooth potential V(r) are obtained, in particular, for the asymptotic coefficients for r to 0 and r to infinity . Comparison with exact solutions (for power potentials) and with a numerical calculation (for a funnel potential) shows that the wave function at the origin is calculated with good accuracy ( approximately 1%). An approximate formula is also obtained for the effective range r/sub 1/ at which a nodeless bound state with orbital angular momentum l arises.

**Author**- Vainberg VM. Mur VD.
**Popov**VS. Sergeev AV. Shcheblykin AV. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- The 1/n expansion in quantum mechanics.
**Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.74, no.3, March 1988, pp.399-411. USSR.
**Abstract**- The classical approximation (l=n-1 to infinity ) for the energy in /sup (0)/ and the semiclassical expansion in quantum mechanics are discussed. A recursive method is proposed for calculating the quantum corrections of arbitrary order to in /sup (0)/, this being valid for both bound and quasistationary states. The generalization of the method to states with an arbitrary number of nodes and the possibility of a more general choice of the parameter of the semiclassical expansion are considered. The method is illustrated by the example of the Yukawa and 'funnel' potentials and for the Stark effect in the hydrogen atom. These examples demonstrate the rapid convergence of the 1/n expansion even for small quantum numbers.

**Author**- Mur VD.
**Popov**VS. **Institution**- Inst. of Exper. & Theor. Phys., USSR.
**Title**- Scaling for the Stark effect in the Rydberg atoms.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.48, no.2, 25 July 1988, pp.67-70. USSR.
**Abstract**- Equations determining the shift and widths of the Rydberg states in a strong electric field (for an arbitrary atom) and the scaling ratios for near-threshold Stark resonances with n/sub 1/ approximately n[right angle bracket][right angle bracket]1, n/sub 2/ and m approximately 1 have been obtained. These scaling ratios have been confirmed experimentally.

**Author**- Vainberg VM. Mur VD.
**Popov**VS. Shcheblykin AV. **Institution**- Int. of Theor. & Exp. Phys., Moscow, USSR.
**Title**- 1/n expansion for the Stark effect in a strong field.
**Source**- Doklady Akademii Nauk SSSR, vol.299, no.4-5, April 1988, pp.1116-20. USSR.
**Abstract**- The 1/n expansion is a promising method with applications in quantum mechanics, field theory, and other areas of theoretical physics. The authors present results of applying this method to the theory of the Stark effect in a strong field using the hydrogen atom as an example.

**Author**- Karnakov VM. Kudryavtsev AE. Mur VD.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Zel'dovich effect in atomic and nuclear physics.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.94, no.7, July 1988, pp.65-79. USSR.
**Abstract**- Restructuring of the atomic spectrum (the Zel'dovich effect) can occur in systems coupled by a Coulomb interaction that is distorted at short distances. The features of this phenomenon are discussed for states with arbitrary angular momentum l. The analytic properties of the solution of the basic equation for this effect in the case hadronic-atom spectra are investigated in the presence of absorption. The conditions under which the spectrum restructuring is replaced by an oscillatory regime are obtained. Some manifestations of the Zel'dovich effect in the physics of hadronic atoms and mesic molecules are considered.

**Accession Number**- 003416864
**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR.
**Editor**- Fodor Z.
**Title**- Zel'dovich effect in atomic and nuclear physics.
**Source**- Eighth Balaton Conference on Nuclear Physics. Intermediate Energy Nuclear Physics. Proceedings of the International Conference on High Energy Nuclear Physics. Central Res. Inst. Phys. 1987, pp.257-65. Budapest, Hungary.
**Conference Information**- Balatonfured, Hungary. 6-11 June 1987.
**Abstract**- In systems bound by the coulomb interaction distorted at short distances there may arise a rearrangement of the atomic spectrum (or the Zel'dovich effect). Specific features of this effect are considered both in the absence and presence of absorption. The Coulomb renormalization of scattering lengths and effective ranges for any l is briefly discussed as well as some examples of the Zel'dovich effect in the physics of hadronic atoms and mesomolecules.

**Author**- Mur VD.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR.
**Title**- Stark effect exhibited by Rydberg atoms and the near-threshold resonances of the photoionization cross section.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.94, no.10, Oct. 1988, pp.125-34. USSR.
**Abstract**- The Stark shifts and the widths of the Rydberg states of atoms in a strong electric field are calculated in the vicinity of E approximately=0. The scaling relationships are obtained for above-barrier resonances. The theoretical results are compared with the experimental data on the photoionization of hydrogen and rubidium in the presence of a static electric field.

**Author**- Mur VD. Pozdnyakov SG.
**Popov**VS. **Title**- 1/n-expansion and calculation of wave functions.
**Source**- Doklady Akademii Nauk SSSR, vol.303, no.4-6, Dec. 1988, pp.1102-7. USSR.
**Abstract**- The possibility of applying the 1/n-expansion to the calculation of not only energies but also wave functions has been discussed in the literature. The authors consider this question in more detail, limiting the analysis to the case of confining potentials, which are of special interest in physics, especially in theories of two- and many-quark systems.

**Author**- Mur VD.
**Popov**VS. Sergeev AV. Shcheblykin AV. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Stark resonances and scaling in Rydberg atoms.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.96, no.1, July 1989, pp.91-106. USSR.
**Abstract**- The Stark shifts and the widths of highly excited states are calculated in a strong electric field (n/sup 4/E approximately 1, n[right angle bracket][right angle bracket]1) for an arbitrary atom (the difference between the atomic field and the Coulomb field is taken into account by introducing quantum defects in a parabolic basis). A correction to the Bohr-Sommerfeld quantization rule, necessitated by the finite barrier penetration factor, is introduced. The possibility of analytic continuation into the above-barrier region is discussed. Scaling relations are obtained via a 1/n-expansion for the near-threshold Stark resonances with quantum numbers n/sub 1/[right angle bracket][right angle bracket]1, n/sub 2/, and m approximately 1. The theoretical results are compared with the experimental data on the photoionization of atoms in the presence of a constant electric field E.

**Author**- Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Experimental Phys., Moscow Eng.-Phys. Inst., USSR.
**Title**- The 1/N expansion in quantum mechanics.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.97, no.1, Jan. 1990, pp.32-46. USSR.
**Abstract**- The authors consider the application of the 1/N expansion to the calculation of energies (of both discrete levels and quasi-stationary states) and wave functions, as well as to the problem of two Coulomb centers and the three-body problem. They show that in many cases (for example, for the cc charmonium states) the method yields high accuracy even for low quantum numbers, including the ground state. The relation of these results to the properties of minimizing-uncertainty coherent states is discussed.

**Author**- Karnakov BM. Mur VD. Pozdnyakov SG.
**Popov**VS. **Institution**- Eng.-Phys. Inst., Moscow, USSR.
**Title**- Analytic structure of the amplitude of dt scattering near the elastic threshold.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.51, no.7, 10 April 1990, pp.352-5. USSR.
**Abstract**- A model-independent description of dt and n alpha scattering has been obtained by means of an effective-radius expansion. The fusion reaction dt to n alpha near the elastic threshold has also been obtained using an effective-radius expansion. The positions of the S-matrix poles, which are associated with the existence of an /sup 5/He*(3/2/sup +/) resonance and with the Coulomb interaction, have been found.

**Author**- Mur VD.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Quantization with allowance for the barrier penetration.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.51, no.10, 25 May 1990, pp.499-502. USSR.
**Abstract**- The correction to the Bohr-Sommerfeld quantization rules, which is associated with the barrier penetration, has been determined. The equation which has been obtained determines the position and width of the quasi-steady-state levels.

**Author**- Mur VD. Pozdnyakov SG.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., State Comm. on Utilization of Atomic Energy, USSR.
**Title**- 1/n expansion and wave functions.
**Source**- Yadernaya Fizika, vol.51, no.2, Feb. 1990, pp.390-6. USSR.
**Abstract**- The authors derive asymptotically exact (for l to infinity ) formulas for wave functions, valid for an arbitrary smooth potential V(r), and also for the asymptotic coefficients at zero and at infinity. They study their accuracy for small values of the quantum numbers. They show that for the potentials used normally in physics the error in the formulas is of the order of few percent even for the ground state (with the exception of shallow levels in short-range potentials).

**Author****Popov**VS. Mur VD. Sergeev AV. Weinberg VM.**Institution**- Int. Centre for Theor. Phys., Trieste, Italy.
**Title**- Strong-field Stark effect: perturbation theory and 1/n expansion.
**Source**- Physics Letters A, vol.149, no.9, 15 Oct. 1990, pp.418-24. Netherlands.
**Abstract**- The Stark shifts and the widths of the ground and excited states of a hydrogen atom are calculated. Two independent calculation methods are used: a summation of divergent perturbation theory series and 1/n expansion. The results of the calculations for the Rydberg (n[right angle bracket][right angle bracket]1) states are in agreement with the experiment.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Inst. Centre for Theor. Phys., Trieste, Italy.
**Title**- 1/n-expansion and scaling for the Stark effect in Rydberg atoms.
**Source**- Physics Letters A, vol.149, no.9, 15 Oct. 1990, pp.425-30. Netherlands.
**Abstract**- The Stark shifts and widths of atomic states in an intense electric field are calculated with the help of the 1/n-expansion. A previous theory is generalized to highly excited states of an arbitrary atom. The scaling relations for near-threshold (E approximately=0) resonances are obtained which are in good agreement with experiment. The modification of the Bohr-Sommerfeld quantization rule due to the finite barrier penetrability is also considered.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., USSR.
**Title**- The 1/n expansion and coherent states.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.97, no.6, June 1990, pp.1729-40. USSR.
**Abstract**- The authors consider states that in the limit n to infinity minimize the uncertainty relation for the radial and transverse (to the plane of the orbit) components of p and r. On going over to classical mechanics such states correspond to closed orbits. This explains the high accuracy of the 1/n expansion even for small quantum numbers, n approximately 1, discovered previously in a number of physical problems.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Inst. of Theor. & Exper. Phys., Moscow, USSR.
**Title**- Quantization rules with barrier penetrability.
**Source**- Journal of the Moscow Physical Society, vol.1, no.1, Feb. 1991, pp.15-31. UK.
**Abstract**- A correction to the quasiclassical quantization rules that allows for the presence of a barrier factor is found. The equation obtained in the process determines both the position of the quasistationary level E/sub r/ and the level width Gamma . The results are compared with numerical solutions of the Schrodinger equation and with those obtained from an exactly solvable model. Finally, a generalization of the Gamow formula for systems with separable variables is derived.

**Author**- Vainberg VM.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- The 1/n expansion for a hydrogen atom in an external field.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.98, no.3, Sept. 1990, pp.847-60. USSR.
**Abstract**- The authors construct a 1/n expansion for a hydrogen atom in constant electric and magnetic fields. The summation of the 1/n expansion enables them with a large degree of accuracy to find the position and the width of atomic levels in a strong field. The method is generalized to resonances in a repulsive Coulomb center and to two-electron atoms.

**Author**- Karnakov BM. Mur VD. Pozdnyakov SG.
**Popov**VS. **Institution**- Inst. of Theor. & Exper. Phys., State Atomic Energy Comm., USSR.
**Title**- Effective-range expansion for dt scattering and the dt to n alpha fusion reaction.
**Source**- Yadernaya Fizika, vol.52, no.6, Dec. 1990, pp.1540-57. USSR.
**Abstract**- Using the effective-range expansion, the authors obtain a model-independent description of dt scattering near the elastic threshold, and also of the dt to n alpha fusion reaction. The nuclear-Coulomb scattering length a/sub cs/ and the effective range r/sub cs/ are obtained from experimental data on the reaction cross section and on the elastic-scattering differential cross section. They study the analytic structure of the S matrix and find the positions of its poles due to the existence of a low-energy resonance /sup 5/He*(/sup 3///sub 2//sup +/) and to the Coulomb interaction. The relationship to various model descriptions of the dt system is discussed.

**Author****Popov**VS. Mur VD. Sergev AV.**Institution**- Inst. of Theoret. & Exp. Phys., Moscow, USSR.
**Title**- Generalization of the Gamow formula to the multidimensional case.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.53, no.9, May 1991, pp.433-6. USSR.
**Abstract**- The Gamow formula for a level width is generalized to the case of multidimensional systems with separable variables. A condition for the applicability of this approximation is found.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Inst. of Theor. & Experimental Phys., Moscow, USSR.
**Title**- Quantization rules for quasistationary states.
**Source**- Physics Letters A, vol.157, no.4-5, 29 July 1991, pp.185-91. Netherlands.
**Abstract**- The modification of the Bohr-Sommerfeld quantization rules, which is due to the barrier penetrability, is found. The equation obtained is valid for an arbitrary analytical potential U(x), obeying the quasiclassical conditions. It determines both the position E/sub r/ and the width Gamma of the quasistationary state. A generalization of the Gamow formula for multidimensional systems with separable coordinates is derived. A comparison with exactly solvable models as well as with numerical solutions of the Schrodinger equation for the Stark problem is performed.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Moscow Eng. Phys. Inst., USSR.
**Title**- Quantization rules with allowance for barrier penetration.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.100, no.1, July 1991, pp.20-44. USSR.
**Abstract**- The semiclassical quantization rules are corrected with allowance for barrier penetration. The resulting equation determines both the position E/sub r/ and the width Gamma of the quasistationary level. A comparison is made with numerical solutions of the Schrodinger equation and with exactly solvable models. The generalization of the Gamov formula for systems with separating variables is found. The possibility of generalizing the results obtained to the relativistic case is discussed.

**Author**- Karnakov BM. Mur VD. Pozdnyakov SG.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR.
**Title**- Poles and resonances in low-energy scattering of charged particles.
**Source**- Yadernaya Fizika, vol.54, no.2, Aug. 1991, pp.400-3. USSR.
**Abstract**- The scattering length and effective range are determined for the system d /sup 3/He, and the positions of the poles of the scattering amplitude in the k plane are calculated. Bounds are obtained for the Coulomb renormalization of the low-energy parameters.

**Author**- Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Generalization of the Gamow formula to the multidimensional case.
**Source**- Yadernaya Fizika, vol.54, no.4, Oct. 1991, pp.950-61. USSR.
**Abstract**- A generalization of the Gamow formula for the width Gamma of a quasistationary level (with energy E=E/sub r/-i Gamma /2) is given for the case of multidimensional systems with separable variables. The condition for applicability of this approximation is obtained, and some examples are considered.

**Author****Popov**VS. Shcheblykin AV.**Institution**- Inst. of Exp. & Theor. Phys., Moscow, USSR.
**Title**- Higher orders of the 1/n expansion and its divergence.
**Source**- Yadernaya Fizika, vol.54, no.6, Dec. 1991, pp.1582-90. Russia.
**Abstract**- A numerical investigation is made of the structure of the higher orders of the 1/n expansion in problems of quantum mechanics (the funnel and Yukawa potentials, and the Stark effect in the hydrogen atom). It is shown that the coefficients epsilon /sup (k)/ of the 1/n expansion have a factorial growth in the limit k to infinity , and their asymptotic form is obtained.

**Author**- Karnakov BM. Mur VD. Pozdnyakov SG.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- On the theory of resonance scattering of low-energy charged particles.
**Source**- Journal of the Moscow Physical Society, vol.2, no.2, May 1992, pp.103-21. UK.
**Abstract**- Using the effective range expansion and new experimental data, the authors calculate the scattering lengths a/sub cs/ and the effective ranges r/sub cs/ for dt-, d/sup 3/He-, and alpha alpha -systems and the positions of the poles of the scattering amplitude in the complex k-plane near the elastic scattering threshold. An expression obtained for r/sub cs/ at the point at which a level arises generalizes the Smorodinskii formula to the case of potentials with Coulomb repulsion at an arbitrary angular momentum l. The authors establish the restrictions on the Coulomb renormalization of low-energy parameters as the charges of the colliding particles increase. Allowing for these restrictions has proved expedient in processing the experimental data.

**Author**- Mur VD. Karnakov BM. Pozdynakov SG.
**Popov**VS. **Institution**- Eng.-Phys., Moscow, Russia.
**Title**- Low-energy scattering parameters for resonance Coulomb systems.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.56, no.3, 10 Aug. 1992, pp.133-6. Russia.
**Abstract**- A formula for the nuclear-Coulomb effective radius r/sub l//sup (sc)/ has been obtained. This formula can be used for an arbitrary angular moment, l and for establishing constraints on the quantity r/sub l//sup (cs)/ which do not depend on the form of the strong potential. The scattering lengths and the effective radii for the lightest hadron systems have been calculated.

**Author****Popov**VS. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Large orders of the 1/n expansion in quantum mechanics.
**Source**- Physics Letters A, vol.172, no.4, 4 Jan. 1993, pp.193-8. Netherlands.
**Abstract**- The asymptotics of large orders of the 1/n expansion in quantum mechanics has been found. It is shown that the coefficients epsilon /sup (k)/ grow as k]a/sup k/ with k to infinity , and the dependence of the parameter a on the coupling constant is investigated.

**Author****Popov**VS. Sergeev AV. Shcheblykin AV.**Institution**- Inst. of Theor. & Exp. Phys., Russia.
**Title**- Structure of higher orders of the 1/n expansion.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.102, no.5, Nov. 1992, pp.1453-68. Russia.
**Abstract**- The asymptote of higher orders of the 1/n expansion in quantum mechanical problems is found. It is shown that the coefficients epsilon /sup (k)/ of the 1/n expansion grow factorially like k] a/sup k/ as k to infinity . The parameter a as a function of the coupling constant is studied. The derived analytic expressions agree with numerical calculations. As examples, the Yukawa, Hulthen and V funnel potentials are considered, as well as the Stark effect in the hydrogen atom and the molecular ion H/sub 2//sup +/.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- On the theory of the above-the-barrier Stark resonances.
**Source**- Physics Letters A, vol.173, no.1, 25 Jan. 1993, pp.63-8. Netherlands.
**Abstract**- A semiclassical 1/n-expansion for the Stark effect in a strong electric field in is considered, which gives simple analytic formulae determining the atomic widths Gamma /sub n/( in ). It has been shown that there is a range in which the widths of the Stark resonances depend almost linearly on the applied electric field, which agrees with numerical calculations.

**Author**- Mur VD. Karnakov BM. Pozdnyakov SG.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Low-energy parameters of the dt and d/sup 3/He systems.
**Source**- Yadernaya Fizika, vol.56, no.2, Feb. 1993, pp.125-44. Russia.
**Abstract**- For resonance Coulomb systems a bound on the effective range r/sub l//sup (cs)/ at the instant of appearance of the l level (a/sub l//sup (cs)/= infinity ) is derived for arbitrary angular momentum. For s states this bound is generalized to the case of finite values of the scattering length. The bound is used to calculate the scattering lengths and effective ranges of the d/sup 3/H and d/sup 3/He systems on the basis of the effective-range expansion and the experimental data on the cross sections for fusion reactions.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- The Stark effect in strong fields: perturbation theory, 1/n-expansion and scaling.
**Source**- Laser Physics, vol.3, no.2, April 1993, pp.462-74. USA.
**Abstract**- The theory of the Stark effect in strong fields that are comparable with an atomic field at the electron orbit is reviewed. The following questions are considered: divergence of the series of the perturbation theory and the methods of their summing (including series with constant signs), 1/n-expansion, the quantum defect, and scaling relationships for near-threshold resonances. Theoretical calculations of Stark shifts and widths of atomic levels, as well as scaling relationships, agree well with the experimental data.

**Author****Popov**VS. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Russian Acad. of Sci., Moscow, Russia.
**Title**- Asymptotic form of higher orders of the 1/n expansion.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.57, no.5, 10 March 1993, pp.273-6. Russia.
**Abstract**- The asymptotic form of higher orders of the 1/n expansion in quantum mechanics is factorial. The Yukawa potential and the hydrogen atom in electric and magnetic fields are discussed.

**Author**- Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- The 1/n-expansion in quantum mechanics and quasi-stationary states.
**Source**- Journal of the Moscow Physical Society, vol.2, no.3, Aug. 1992, pp.189-217. UK.
**Abstract**- Applications of the 1/n-expansion to energy and wavefunction calculations, especially for quasi-stationary states are considered. As an illustration the authors have examined the funnel and Yukawa potentials, the Stark effect in a strong field and a hydrogen atom in electric and magnetic fields. They have found that in many cases the method ensures high accuracy, even for small quantum number, n approximately 1. The connection between these results and the properties of coherent states is briefly discussed. Finally, the behaviour of the large-order terms in the 1/n-expansion is investigated.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- Quantization rules for above-barrier resonances.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.57, no.7, 10 April 1993, pp.406-9. Russia.
**Abstract**- The Bohr-Sommerfeld quantization condition is analytically continued into the above-barrier region. The result is used to find the asymptotic behavior of the energies of resonances under strong-coupling conditions.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- The WKB method for resonances.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.104, no.1, July 1993, pp.2293-313. Russia.
**Abstract**- The authors establish the analytic continuation of the Bohr-Sommerfeld quantization rule for potentials with a barrier into the range of above-the-barrier resonances. The derived equations are valid for an arbitrary analytic potential satisfying quasi-classical conditions and determine both the position E/sub r/ and width Gamma of the resonance. The results are illustrated by a number of model potentials and also for the Stark effect in a strong field. They find the asymptotic behavior of the resonance energy in the tight-binding approximation.

**Author**- Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- Quantization rules for above-the-barrier resonances.
**Source**- Journal of the Moscow Physical Society, vol.3, no.1, Feb. 1993, pp.23-36. UK.
**Abstract**- This paper establishes the analytic continuation of the Bohr-Sommerfeld quantization rule in the range of above-the-barrier resonances. The derived equation is valid for an arbitrary analytic potential satisfying quasi-classical conditions and determines both the position of the resonance E/sub r/ and resonance's width Gamma . The results are illustrated by model potentials, and the asymptotic behaviour of the resonance energy in the strong-coupling approximation is established.

**Author**- Alliluev SP.
**Popov**VS. **Institution**- Moscow Physical-Tech. Inst., Dolgoprudnyi, Russia.
**Title**- On the theory of the stark effect: the influence of dimensionality.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.104, no.5, Nov. 1993, pp.3569-78. Russia.
**Abstract**- A theory of the Stark effect in the D-dimensional "hydrogen atom" is developed. Recursion relations determining the higher-order perturbation terms E/sub k/ for the ground state are obtained, and the behavior of the coefficients E/sub k/ as k is increased, as well as their asymptote as k to infinity , are investigated. Analogous problems for the D-dimensional funnel potential are also considered.

**Author**- Mur VD.
**Popov**VS. Simonov YaA. Yurov VP. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Description of relativistic heavy-light quark-antiquark systems via Dirac equation.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.105, no.1, Jan. 1994, pp.3-27. Russia.
**Abstract**- Starting from the QCD Lagrangian and taking into account both perturbative and nonperturbative effects, we use the method of vacuum correlators to derive the Dirac equation (rigorously for the Coulomb interaction and heuristically for the confining potential) for a system consisting of a light quark and heavy antiquark. As a result the confining potential is a Lorentz scalar, and the Coulomb part the fourth component of a 4-vector. The energy spectrum of the Dirac equation is considered for these potentials. Numerical calculations of energy eigenvalues E=E/sub nk/ are performed, and some exact solutions of the Dirac equation in the case E=0 are found. An effective-potential method convenient for qualitative study of the solutions of the Dirac equation is developed. The connection with experimental spectra of D- and B-mesons is briefly discussed.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Critical fields and above-barrier Stark resonances.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.59, no.3, 10 Feb. 1994, pp.150-3. Russia.
**Abstract**- Exact values of the critical field C/sub c/ are calculated for a hydrogen atom, including the case of the ground state. The widths Gamma /sub n/ of Stark resonances at C=C/sub c/, are also calculated. In the above-barrier region (C[right angle bracket]C/sub c/ the widths Gamma /sub n/(C) are essentially linear functions of the electric field strength.

**Author****Popov**VS. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Asymptotic form of higher orders of the 1/n expansion for multidimensional problems.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.105, no.3, March 1994, pp.568-91. Russia.
**Abstract**- The asymptotic form of higher orders of the 1/n expansion is analyzed for multidimensional problems of quantum mechanics and atomic physics. Problems in which variables can and cannot be separated are discussed. Examples of the former are the molecular hydrogen atom and the Stark effect; an example of the latter is the hydrogen atom in electric and magnetic fields. The asymptotic form is always factorial. Its parameters can be calculated through a calculation of electron tunneling trajectories by the "imaginary time" method. Possible types of singularities of the asymptotic parameter alpha at the collision point of the classical solutions are determined.

**Author**- Mur VD. Karnakov BM. Pozdnyakov SG.
**Popov**VS. **Institution**- Moscow Inst. of Eng. Phys., Russia.
**Title**- n alpha scattering in the vicinity of the /sup 5/He/sup */ resonance.
**Source**- Yadernaya Fizika, vol.57, no.5, May 1994, pp.820-5. Russia.
**Abstract**- The amplitude of n alpha scattering in the vicinity of the /sup 5/He* (3/2/sup +/) resonance is expressed in terms of the scattering length and effective range of the dt system. The total cross section of scattering near the threshold of the dt channel is compared with experimental data. It is shown that the phase shift analysis of elastic n alpha scattering at these energies is an effective method for choosing the low-energy parameters of the dt system.

**Author****Popov**VS. Sergeev AV. Mur VD. Shcheblykin AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- On the asymptotics of high-order terms of the 1/n expansion.
**Source**- Yadernaya Fizika, vol.57, no.6, June 1994, pp.1119-28. Russia.
**Abstract**- Analytic and numerical methods for determining the asymptotics of high-order terms of the 1/n expansion in quantum-mechanical problems are developed. It is shown that this asymptotics is always of the factorial type. The dependence of parameters of the asymptotics on the form of the potential and on the coupling constant is especially analyzed in the vicinity of the point of collision of classical solutions.

**Author****Popov**VS. Mur VD.**Institution**- Inst. of Theor. & Experimental Phys., Moscow, Russia.
**Title**- Perturbation theory for quasistationary levels.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.60, no.1, 10 July 1994, pp.64-7. Russia.
**Abstract**- A perturbation theory formula is derived for the energies of quasistationary states (resonances) in the semiclassical approximation. This formula is valid for resonances either below or above the barrier. It is illustrated for several potentials, for which a comparison can be made with exact solutions.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Critical electric fields and Stark resonances in the hydrogen atom.
**Source**- Physics Letters A, vol.193, no.2, 26 Sept. 1994, pp.159-64. Netherlands.
**Abstract**- The exact values of critical electric fields E/sub c/ for different states of a hydrogen atom, including its ground state, as well as the widths Gamma /sub n/ of the Stark resonances at E=E/sub c/, are calculated (at E=E/sub c/ (n/sub 1/, n/sub 2/, m) the potential barrier for an electron in the (n/sub 1/, n/sub 2/, m) state disappears). Using the modified quantization condition with barrier penetrability included, we explain the dependence of Gamma /sub n/ on the principal quantum number n.

**Author****Popov**VS. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Large orders of 1/n-expansion for multidimensional problems.
**Source**- Physics Letters A, vol.193, no.2, 26 Sept. 1994, pp.165-72. Netherlands.
**Abstract**- The asymptotics of large orders of the 1/n-expansion is investigated for multidimensional problems of quantum mechanics and atomic physics, including those with separable variables (the hydrogen molecular ion H/sub 2//sup +/), and those where separation of variables is impossible (a hydrogen atom in electric and magnetic fields). It is shown that the parameters of the asymptotics can be found by means of calculating sub-barrier trajectories with the help of the "imaginary time" method, as well as by solution of the eikonal equation.

**Author****Popov**VS. Mur VD. Sergeev AV.**Institution**- Int. Inst. of Phys., Moscow, Russia.
**Title**- Theory of the Stark effect in a strong field: critical fields, above-barrier resonances, dependence on dimensionality.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.106, no.4, Oct. 1994, pp.1001-30. Russia.
**Abstract**- We have calculated accurate values of the critical fields E/sub c/ for various states (n/sub 1/, n/sub 2/, m) of a hydrogen atom, including the ground state. Using modified quantization rules that take into account the finite permeability of the barrier, we determine how the widths Gamma /sub n/ of the Stark resonances depend on n for E=E/sub c/. Using the 1/n expansion, along with several model problems as examples, we show that in the above-barrier (E[right angle bracket]E/sub c/) region the Stark widths Gamma /sub n/(E) depend almost linearly on the electric field. We determine the dependence of the ground-state Stark shift of a d-dimensional "hydrogen atom" on the dimensionality D of the space.

**Author**- Karnakov BN. Mur VD.
**Popov**VS. **Institution**- Eng. Phys. Inst., Moscow, Russia.
**Title**- 1/n expansion for wave functions.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.106, no.4, Oct. 1994, pp.976-92. Russia.
**Abstract**- Using the 1/n expansion, analytical formulas for wave functions (and in particular for the asymptotic coefficients c/sub lp/ at zero) in an arbitrary potential V(r) are obtained, valid in both the classically permitted and sub-barrier regions. Comparison with the exact solutions and numerical calculations for power and short-range potentials shows that the region of applicability of the formulas can be stretched to small quantum numbers n approximately 1 (for states with p=0 and 1, where p is the number of nodes in the radial wave function). With increasing p the accuracy of the formulas decreases, but in this case the WKB method may be applied, and as a result psi (0) for states with arbitrary quantum numbers can be calculated. When using WKB, a generalization of the turning-point matching conditions for quasiclassical wave functions is essential.

**Author**- Kamakov BM. Mur VD.
**Popov**VS. **Institution**- Eng.-Phys. Inst., Moscow, Russia.
**Title**- 1/n expansion for wave functions.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.107, no.2, Feb. 1995, pp.976-1218. Russia.
**Abstract**- Using the 1/n expansion, analytical formulas for wave functions (and in particular for the asymptotic coefficients c/sub lp/ at zero) in an arbitrary potential V(r) are obtained, valid in both the classically permitted and sub-barrier regions. Comparison with the exact solutions and numerical calculations for power and short-range potentials shows that the region of applicability of the formulas can be stretched to small quantum numbers n approximately 1 (for states with p=0 and 1, where p is the number of nodes in the radial wave function). With increasing p the accuracy of the formulas decreases, but in this case the WKB method may be applied, and as a result psi (0) for states with arbitrary quantum numbers can be calculated. When using WKB, a generalization of the turning-point matching conditions for quasiclassical wave functions is essential.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Inter. Phys. Inst., Moscow, Russia.
**Title**- The matching conditions in the WKB method.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.107, no.6, June 1995, pp.1768-79. Russia.
**Abstract**- The modified matching condition for quasiclassical wave functions on the two sides of a turning point has been obtained for the radial Schrodinger equation. It differs essentially from the usual Kramers condition which holds for one-dimensional problems. Comparison with exact and numerical solutions of the Schrodinger equation shows that the modified matching condition not only makes the quasiclassical approximation in the subbarrier region asymptotically exact in the limit n to infinity , but also considerably enhances its accuracy even in the case of small quantum numbers, n approximately 1. Power-law and short-range potentials are considered in detail. It is shown that for zero-energy states in a short-range attractive potential (i.e., at the point when the bound l-state occurs, l[right angle bracket]or=1) the normalization condition generally used in the WKB method should also be modified.

**Author**- Mur VD.
**Popov**VS. **Institution**- State Res. Center, Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Perturbation theory for quasistationary states.
**Source**- Yadernaya Fizika, vol.58, no.8, Aug. 1995, pp.1413-25. Russia.
**Abstract**- Perturbation theory (to the first order) for quasistationary states is developed in the semiclassical approximation. Formulas for the energy shift delta E/sub r/ of a resonance, as well as for the variation of its width delta Gamma due to a perturbative potential delta U, are obtained. These formulas are valid both for subbarrier and abovebarrier resonances. The results are illustrated using a number of potentials for which semiclassical expressions can be compared with exact solutions of the Schrodinger equation. The proposed scheme is generalized to a multidimensional case with completely separating variables.

**Author**- Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Coulomb corrections to nuclear scattering lengths and effective ranges for weakly bound systems.
**Source**- Yadernaya Fizika, vol.59, no.1, Jan. 1996, pp.68-77. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- A procedure is considered for extracting the purely nuclear scattering length a/sub s/ and effective range r/sub s/ (which correspond to a strong-interaction potential V/sub s/ with disregarded Coulomb interaction) from the experimentally determined nuclear quantities a/sub cs/ and r/sub cs/ which are modified by Coulomb interaction. The Coulomb renormalization of a/sub s/ and r/sub s/ is especially strong if the system under study involves a level with energy close to zero (on the nuclear scale). This applies to formulas that determine the Coulomb renormalization of the low energy parameters of s scattering (l=0). Detailed numerical calculations are performed for coefficients appearing in the equations that determine Coulomb corrections for various models of the potential V/sub s/(r). This makes it possible to draw qualitative conclusions about the dependence of Coulomb corrections on the form of the strong-interaction potential and, in particular, on its small-distance behavior. A considerable enhancement of Coulomb corrections to the effective range r/sub s/ is found for potentials with a barrier.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- On matching conditions in the WKB method.
**Source**- Physics Letters A, vol.210, no.6, 22 Jan. 1996, pp.402-8. Publisher: Elsevier, Netherlands.
**Abstract**- The modified matching conditions for quasiclassical wave functions on both sides of a turning point for the radial Schroedinger equation have been obtained. They differ significantly from the usual Kramers condition which holds for the one-dimensional case. Namely, the ratio C/sub 2//C/sub 1/ in the subbarrier and the classical allowed regions is not a universal constant (C/sub 2//C/sub 1/=1/2, as usual), but depends on the values of the orbital angular momentum l, energy E and on the behaviour of the potential V(r) at r to 0. The comparison with exact and numerical solutions of the Schroedinger equation shows that the modified matching conditions not only make the quasiclassical approximation in the subbarrier region asymptotically exact within the n to infinity limit, but also considerably enhances its accuracy even in the case of small quantum numbers, n~1. The power-law, funnel and short-range potentials are considered in detail.

**Author****Popov**VS. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Effect of a magnetic field on the ionization of atoms.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.63, no.6, 25 March 1996, pp.398-402. Publisher: AIP, Russia.
**Abstract**- The ionization probability of an atomic s state under the action of static electric and magnetic fields is calculated taking into account the Coulomb interaction between the escaping electron and the atomic core. The structure of the perturbation series for the energy of the level is investigated and the asymptotic behavior of the higher orders of the perturbation theory is found.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- Two comments on the WKB method.
**Source**- Doklady Akademii Nauk, vol.347, no.4-6, April 1996, pp.466-8. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- The WKB method, or quasi-classical approximation, is one of the most important approximate methods in quantum mechanics and mathematical physics. The critical element of the method is the sewing conditions for the quasi-classical wave functions on different sides of a turning point x=a. In the one-dimensional case and for smooth potentials, the Kramers boundary conditions are usually satisfied. We prove these boundary conditions, in general, to be invalid for the radial wave functions: the ratio C"/C is not a universal constant but depends on the orbital momentum l, the behavior of the potential at small distances, and the energy of state.

**Author**- Mur VD.
**Popov**VS. Sergeev AV. **Institution**- Eng. Phys. Inst., Acad. of Sci., Moscow, Russia.
**Title**- Coulomb corrections to scattering lengths and effective ranges for l not=0.
**Source**- Yadernaya Fizika, vol.59, no.11, Nov. 1996, pp.1958-65. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- Coulomb corrections to the low-energy scattering parameters (scattering lengths and effective ranges) of states with nonzero orbital angular momentum are studied. The greatest attention is paid to the case of the p wave. Resonance scattering is considered for a system with a shallow nuclear bound or quasistationary (resonance) p level. Coulomb corrections are calculated numerically, and their dependence on the shape of the strong-interaction potential-in particular, on its short-distance behavior-is studied in detail.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. fur Theor. Phys., Bern Univ., Switzerland.
**Title**- 1/n-expansion for asymptotic coefficients of radial wave functions in quantum mechanics.
**Source**- Physics Letters A, vol.224, no.1-2, 30 Dec. 1996, pp.15-21. Publisher: Elsevier, Netherlands.
**Abstract**- Using the 1/n-expansion, we obtain analytic formulae for the bound state radial wave functions, including its asymptotic coefficients at r to 0 and r to infinity , for an arbitrary smooth potential V(r). The formulae are asymptotically exact in the limit n/sub r/ to infinity (n=n/sub r/+l+1 is the principal quantum number and the expansion parameter is 1/n). Comparison with exact solutions and numerical calculations for the power-law and short-range potentials show that the applicability region of these formulae is usually prolonged up to small quantum numbers, n~1. With growing n/sub r/, the accuracy of the formulae decreases, but the WKB method becomes applicable in this case.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Model-independent description of the dt and d/sup 3/He systems near low-energy resonances.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.110, no.5, Nov. 1996, pp.1537-56. Publisher: AIP, Russia.
**Abstract**- An expansion of the effective radius is employed in a model-independent description of the dt and d/sup 3/He systems in the vicinity of the low-energy /sup 5/He*(3/2/sup +/) and /sup 5/Li*(3/2/sup +/) resonances. The Coulomb-nuclear scattering lengths (a/sub cs/) and effective radii (r/sub cs/) for states with orbital angular momentum l=0, as well as the astrophysical function s(E), are extracted from existing experimental data for the cross sections of the nuclear fusion reactions dt to n alpha and d/sup 3/He to p alpha in the vicinity of the resonances (data on elastic dt and n alpha scattering are also employed in the case of the dt system). Extensive use is made of the generalization of the Schwinger-Smorodinskii equation for the effective radius to the case of potentials with a Coulomb barrier. A bound is established on the value of the Coulomb-nuclear effective radius r/sub l//sup (cs)/ with an arbitrary value of the orbital angular momentum l, which does not depend on the specific form of the strong potential V/sub s/(r). Numerical calculations of the form factors P and Q appearing in the expansion of the effective radius are performed for different models of V/sub s/(r), and the problem of the stability of the results obtained toward possible variations of the form of the strong interaction is discussed. The analytic structure of the scattering amplitude near the elastic threshold is investigated in the presence of absorption, i.e., open reaction channels, in the system. Two series of Coulomb poles in the complex k plane, which converge near the elastic threshold (k=0), are found for the dt scattering amplitude.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Quasiclassical theory of atomic ionization in electric and magnetic fields.
**Source**- Physics Letters A, vol.229, no.5, 26 May 1997, pp.306-12. Publisher: Elsevier, Netherlands.
**Abstract**- Using the "imaginary time" method we have calculated (in the quasiclassical approximation) the probability of ionization of the atomic s-state in static electric and magnetic fields. The Coulomb interaction between the emitted electron and the atomic remainder is taken into account. The results obtained are valid for external fields E and H which are smaller than characteristic atomic fields. The case of mutually orthogonal fields (the Lorentz ionization) is carefully studied.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Eng. Phys. Inst., Moscow Tech. Univ., Russia.
**Title**- Contribution to the theory of Lorentzian ionization.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.65, no.5, 10 March 1997, pp.391-6. Publisher: AIP, Russia.
**Abstract**- The probability w/sub L/ of Lorentzian ionization, which arises when an atom or ion moves in a constant magnetic field, is calculated in the quasiclassical approximation. The nonrelativistic (v[left angle bracket]or approximately=e/sup 2//h(cross)=1, v is the velocity of the atom) and ultrarelativistic (v to c=137) cases are examined and the stabilization factor S, which takes account of the effect of the magnetic field on tunneling of an electron, is found.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- A model-independent description of dt and d/sup 3/He systems near low-energy resonances.
**Source**- Baltzer. Hyperfine Interactions, vol.101-102, Oct. 1996, pp.401-11. Netherlands.
**Conference Information**- International Symposium on Muon Catalyzed Fusion and the Physics of Exotic Atoms and Molecules ( mu CF-95). Dubna, Russia. Joint Inst. Nucl. Res. Russian Minist. Sci. & Tech. Policy. et al. 19-24 June 1995.
**Abstract**- The effective range expansion is used to obtain a model-independent description of dt and d/sup 3/He systems in the vicinity of the low-energy resonances /sup 5/He*(3/2/sup +/) and /sup 5/Li*(3/2/sup +/). The Coulomb-modified nuclear scattering lengths alpha /sub cs/ and effective ranges r/sub cs/ with angular momentum l=0 as well as the astrophysical function s(E) are calculated from the modern experimental data for cross sections of the dt to n alpha and d/sup 3/He to p alpha fusion reactions in the vicinity of the resonances. To this end the generalization of the Smorodinsky formula for effective range to the case of potentials with the Coulomb barrier, as well as the phase analysis of the elastic n alpha -scattering near the resonance /sup 5/He*(3/2/sup +/), are essentially used. The analytical structure of the scattering amplitudes in presence of absorption in the system is studied. Two series of the Coulomb poles of the dt-scattering amplitude in the complex k-plane, condensing to the elastic threshold, are found.

**Author****Popov**VS. Mur VD. Karnakov BM.**Institution**- Inst. of Theor. & Exp. Phys., Acad. of Sci., Moscow, Russia.
**Title**- The imaginary-time method for relativistic problems.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.66, no.4, 25 Aug. 1997, pp.213-18. Publisher: AIP, Russia.
**Abstract**- A relativistic version of the imaginary-time method is presented. The method is used to calculate the probability w of ionization of a bound state by electric and magnetic fields of various configurations (including the case when the binding energy E/sub b/ is comparable to mc/sup 2/). The formulas cover as limiting cases both the ionization of nonrelativistic bound systems (atoms and ions) and the case E/sub b/=2mc/sup 2/, when w equals the probability of electron-positron pair production from the vacuum in the presence of a strong field.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Moscow State Eng. Phys. Inst., Moscow, Russia.
**Title**- Quasiclassical theory of atomic ionization in electric and magnetic fields.
**Source**- Journal of the Moscow Physical Society, vol.7, no.3, 1997, pp.237-59. Publisher: Allerton Press, USA.
**Abstract**- Using the "imaginary time" method, we have calculated (in the quasi-classical approximation) the probability of ionization of an atomic s-state under the action of static electric and magnetic fields, taking account of the Coulomb interaction between the emitted electron and the atomic core. The results obtained are valid for arbitrary fields E and H that are small compared to the characteristic atomic fields. Special attention is paid to the case of mutually orthogonal fields (the Lorentz ionization). The dependence of the ionization probability on the spatial dimension D is briefly discussed. The formulas for the momentum and energy distributions of the emitted electrons for ionization of a negative ion are obtained.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- State Res. Center, Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Langer transformation and modified matching conditions in the WKB method.
**Source**- Yadernaya Fizika, vol.61, no.3, March 1998, pp.481-93. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- It is shown that the Kramers-Langer matching condition in the WKB method is not asymptotically exact in calculating wave functions in the subbarrier region if the orbital angular momentum is l~1. Modified matching conditions free from this drawback are obtained; they depend on l, on the energy E, and on the small-distance behavior of the potential V(r). The precision of the new matching conditions is studied in calculating the asymptotic coefficients in the wave function at the origin and at infinity. Some regularities in the behavior of the semiclassical spectrum for power-law potentials are considered.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Ionization of atoms in electric and magnetic fields and the imaginary time method.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.113, no.5, May 1998, pp.1579-605. Publisher: AIP, Russia.
**Abstract**- A semiclassical theory is developed for the ionization of atoms and negative ions in constant, uniform electric and magnetic fields, including the Coulomb interaction between the electron and the atomic core during tunneling. The case of crossed fields (Lorentz ionization) is examined specially, as well as the limit of a strong magnetic field. Analytic equations are derived for arbitrary fields E and H that are weak compared to the characteristic intraatomic fields. The major results of this paper are obtained using the "imaginary time" method (ITM), in which tunneling is described using the classical equations of motion but with purely imaginary "time." The possibility of generalizing the ITM to the relativistic case, as well as to states with nonzero angular momentum, is pointed out.

**Author****Popov**VS. Sergeev AV.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Ionization of atoms in weak fields and the asymptotic behavior of higher-order perturbation theory.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.113, no.6, June 1998, pp.2047-55. Publisher: AIP, Russia.
**Abstract**- Using the imaginary time method, we study the structure of the perturbation series for the hydrogen atom in electric E and magnetic H fields. It is shown that there is a "critical" value of the ratio gamma =H/E at which the perturbation series for the ground state changes from having a fixed sign (for gamma [left angle bracket] gamma /sub c/) to having a variable sign (for gamma [right angle bracket] gamma /sub c/). This conclusion is confirmed by direct higher-order perturbation calculations. The change in the asymptotic regime is explained by competition among the contributions of the various complex trajectories that describe the subbarrier motion of the electrons. Here the parameter gamma /sub c/ depends on the angle theta between the electric and magnetic fields.

**Author**- Mur VD. Karnakov BM.
**Popov**VS. **Institution**- Eng. Phys. Inst., Moscow, Russia.
**Title**- Relativistic version of the imaginary-time formalism.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.114, no.3, Sept. 1998, pp.798-820. Publisher: AIP, Russia.
**Abstract**- A relativistic version of the quasiclassical imaginary-time formalism is developed. It permits calculation of the tunneling probability of relativistic particles through potential barriers, including barriers lacking spherical symmetry. Application of the imaginary-time formalism to concrete problems calls for finding subbarrier trajectories which are solutions of the classical equations of motion, but with an imaginary time (and thus cannot be realized in classical mechanics). The ionization probability of an s level, whose binding energy can be of the order of the rest energy, under the action of electric and magnetic fields of different configuration is calculated using the imaginary-time formalism. Besides the exponential factor, the Coulomb and pre-exponential factors in the ionization probability are calculated. The Hamiltonian approach to the tunneling of relativistic particles is described briefly. Scrutiny of the ionization of heavy atoms by an electric field provides an additional argument against the existence of the "Unruh effect".

**Author****Popov**VS. Mur VD. Karnakov BM.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Relativistic version of the imaginary time method.
**Source**- Physics Letters A, vol.250, no.1-3, 21 Dec. 1998, pp.20-4. Publisher: Elsevier, Netherlands.
**Abstract**- We present the relativistic version of the imaginary time method (ITM). The ionization probability w of the bound state under the action of electric and magnetic fields (for the case when the binding energy E/sub b/ is comparable with mc/sup 2/) has been calculated using the ITM. The formulae obtained cover both the ionization of nonrelativistic bound systems (atoms, ions) and the case of E/sub b/=2mc/sup 2/, when w is compared with the probability of electron-positron pair production from vacuum in a strong field.

**Author**- Mur VD. Karnakov BM.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- On the asymptotic coefficient for atoms and ions.
**Source**- Doklady Akademii Nauk, vol.365, no.1-3, March 1999, pp.329-32. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- An s-wave function has asymptotic behavior. The asymptotic coefficients C are important physical parameters of the bound state and are widely used in quantum mechanics, atomic and nuclear physics, the inverse problem of quantummechanical scattering theory, etc. Exact calculation of these coefficients is a fairly complicated problem; for example, in the case of heavy atoms, a numerical solution of the Hartree-Fock equations for many-electron atoms is required, with the computational errors being as large as 10-30% or more in some cases. We give simple analytic approximations, which correspond closely to numerical calculations, for both the coefficients C and the moments of distribution of valence s-electron density.

**Author****Popov**VS. Karnakov BM. Mur VD.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Lorentz ionization of atoms in a strong magnetic field.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.115, no.5, May 1999, pp.1641-63. Publisher: AIP, Russia.
**Abstract**- Lorentz ionization emerges due to the motion of atoms or ions in a strong magnetic field. We use the semiclassical approximation to calculate the probability w/sub L/ of Lorentz ionization. We also find the stabilization factor S, which takes into account the reduction by the magnetic field of the probability of ionization decay of the bound s state. We estimate the probabilities w/sub L/ in magnetic-cumulation experiments and in astrophysics. We also qualitatively examine the dynamics of the magnetic cumulation process with allowance for the conductivity of the shell. Finally, we discuss a paradox related to the use of the quasistationary solution at the shell expansion stage.

**Author**- Karnakov BM. Kulik AV. Mur VD.
**Popov**VS. **Institution**- Eng. Phys. Inst., Moscow, Russia.
**Title**- Effective range expansion and low-energy parameters for dt, d/sup 3/He, and alpha alpha systems.
**Source**- Journal of the Moscow Physical Society, vol.8, no.4, 1998, pp.291-315. Publisher: Allerton Press, USA.
**Abstract**- An effective range expansion is used to obtain a model-independent description of dt and d/sup 3/He systems in the vicinity of the low-energy resonances /sup 5/He*(3/2/sup +/) and /sup 5/Li*(3/2/sup +/). The Coulomb modified nuclear scattering lengths a/sub cs/ and effective ranges r/sub cs/ with angular momentum l=0, as well as the astrophysical functions s(E), are calculated using modern experimental data for the cross sections of the dt to n alpha and d/sup 3/He to p alpha fusion reactions in the vicinity of the resonances. A the generalization of the Schwinger-Smorodinsky formula for the effective range to the case of potentials with a Coulomb barrier, as well as a phase analysis of elastic n alpha scattering near the /sup 5/He*(3/2/sup +/) resonance, are used. An analogous approach has been developed for an alpha alpha system, but in this case it appeared to be necessary to also use specific models for the strong potential. The values of a/sub cs/, r/sub cs/, and the shape parameters P/sub cs/ and Q/sub cs/ for the alpha alpha system, as well as the /sup 8/Be(0/sup +/) resonance width, have been calculated. The analytical structure of the scattering amplitudes in the presence of absorption in the system is studied. Two series of the Coulomb poles of the dt scattering amplitude in the complex k plane that condense to the elastic threshold k=0 are found.

**Author**- Mur VD. Karnakov BM.
**Popov**VS. **Institution**- Eng. Phys. Inst., Moscow, Russia.
**Title**- Asymptotic coefficients for atoms and ions.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.115, no.2, Feb. 1999, pp.521-41. Publisher: AIP, Russia.
**Abstract**- We derive convenient analytical formulas in the effective-range approximation for the asymptotic coefficient C/sub kappa / of the radial wave function at infinity and for the average radius of the system. A comparison with the results of numerical calculations (done by the Hartree-Fock method) for multi-electron atoms and ions reveals that this approximation has good accuracy for valence s-electrons in all atoms from hydrogen to uranium. We calculate the values of the scattering lengths and the effective ranges for electron-atom and electron-ion scattering. We also examine the quasiclassical approximation for C/sub kappa /. Finally, we discuss the logarithmic increase in the effective ranges of ns states as n to infinity.

**Author**- Kamakov BM. Mur VD.
**Popov**VS. **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- Quasiclassical approximation with the centrifugal potential excluded.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.116, no.2, Aug. 1999, pp.511-25. Publisher: AIP, Russia.
**Abstract**- We develop a modification of the WKB method (the modified quantization method, or MQM) for finding the radial wave functions. The method is based on excluding the centrifugal potential from the quasiclassical momentum and changing correspondingly the phase in the Bohr-Sommerfeld quantization condition. MQM is used to calculate the asymptotic coefficients at zero and at infinity. We use the examples of power-law and funnel potentials to show that MQM not only dramatically broadens the possibilities of studying the energy spectrum and the wave functions analytically but also ensures accuracy to within a few percent even when one calculates states with a radial quantum number n/sub r/~1, provided that the angular momentum l is not too large. We also briefly discuss the possibility of generalizing MQM to the relativistic case (the spinless Salpeter equation).

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Moscow State Eng. Phys. Inst., Russia.
**Title**- Hamiltonian approach to the problem of tunneling of relativistic particles.
**Source**- Yadernaya Fizika, vol.62, no.8, Aug. 1999, pp.1444-50. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- The Hamiltonian approach to the problem of tunneling of relativistic particles has been developed. The method is described by applying it to a specific example, the theory of the ionization of a relativistic bound state (whose binding energy is comparable with the electron rest energy mc/sup 2/) under the effect of external electric and magnetic fields. It has been shown that the Hamiltonian approach makes it possible to obtain an explicit expression for both exponential and preexponential factors in the ionization probability.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Energy and angular distributions of photoelectrons in multiphoton ionization.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.70, no.8, 25 Oct. 1999, pp.493-7. Publisher: AIP, Russia.
**Abstract**- The energy and angular distributions of photoelectrons in the photoionization of an atom or ion by powerful laser radiation are calculated. The results obtained are valid for all values of the Keldysh parameter gamma . Linear and circular polarizations of the light are considered.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Tunneling and above-barrier ionization of atoms in a laser radiation field.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.118, no.1, July 2000, pp.56-76. Publisher: MAIK Nauka, Russia.
**Abstract**- Calculations are made of the energy and angular distributions of photoelectrons during tunneling ionization of an atom or an ion under the action of high-power laser radiation (for all values of the Keldysh parameter gamma ). Cases of linear, circular, and elliptic polarizations of the electromagnetic wave are considered. The probability of above-barrier ionization of hydrogen atoms in a low-frequency ( gamma [left angle bracket][left angle bracket]1) laser field is calculated. Formulas are given for the momentum spectrum of the electrons when an atomic level is ionized by a general type of alternating electric field (for the case of linear polarization). An analysis is made of tunneling interference in the energy spectrum of the photoelectrons. Analytic approximations are discussed for the asymptotic coefficient C/sub k/ of the atomic wave function at infinity (for s-wave electrons).

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- On the theory of tunneling and above-barrier ionization of atoms and ions in a strong laser field.
**Source**- Laser Physics, vol.10, no.5, Sept.-Oct. 2000, pp.1033-46. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- Energy and angular distributions of photoelectrons emerging from the tunneling ionization of an atom or an ion under the action of high-power laser radiation (for all the values of the Keldysh parameter gamma ) are calculated. The cases of linearly, circularly, and elliptically polarized electromagnetic waves are considered. The probability of above-barrier ionization of hydrogen atoms in a low-frequency ( gamma [left angle bracket][left angle bracket]1) laser field is calculated. Formulas for the momentum spectrum of electrons in the case when an atomic level is ionized with an AC (linearly polarized) electric field of general form are presented. The method of "imaginary time" is applied to analyze ionization of atoms in a nonmonochromatic field and the Coulomb correction in the range where gamma [left angle bracket]or approximately=1. Analytical approximations for the asymptotic coefficient C/sub K/ of the atomic wave function at infinity (for s electrons) are discussed.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Multiphoton ionization of atoms by an ultrashort laser pulse.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.73, no.1, 2001, pp.3-7. Publisher: MAIK Nauka, Russia.
**Abstract**- Closed analytical expressions for the probability of multiphoton ionization of atoms and ions by a time-varying electric field E(t) are obtained by the imaginary time method. These expressions apply for arbitrary values of the Keldysh parameter gamma . The dependence of the ionization probability and the photoelectron momentum spectrum on the shape of an ultrashort laser pulse is considered.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Critical charge in quantum electrodynamics.
**Source**- Yadernaya Fizika, vol.64, no.3, March 2001, pp.421-46. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- The critical nuclear charge Z/sub cr/ and the critical distance R/sub cr/ in the system of two colliding heavy nuclei-they are defined as those at which the ground-state level of the electron spectrum descends to the boundary of the lower continuum, with the result that beyond them (that is, for Z[right angle bracket]Z/sub cr/ or R[left angle bracket]R/sub cr/) spontaneous positron production from a vacuum becomes possible-are important parameters in the quantum electrodynamics of ultrastrong Coulomb fields. Various methods for calculating Z/sub cr/ and R/sub cr/ are considered, along with the dependence of these quantities on the screening of the Coulomb field of a nucleus by the electron shell of the atom, on an external magnetic field, on the particle mass and spin, and on some other parameters of relevance. The effective-potential method for the Dirac equation and the application of the Wentzel-Kramers-Brillouin method to the Coulomb field for Z[right angle bracket]137 and to the two-body Salpeter equation for the quark-antiquark system are discussed. Some technical details in the procedure for calculating the critical distance R/sub cr/ in the relativistic problem of two Coulomb centers are described.

**Author**- Karnakov BM. Mur VD.
**Popov**VS. **Institution**- Moscow State Eng. Phys. Inst., Russia.
**Title**- Semiclassical approximation and 1/n expansion in quantum-mechanical problems.
**Source**- Yadernaya Fizika, vol.64, no.4, April 2001, pp.729-49. Publisher: MAIK Nauka/Interperiodica Publishing, Russia.
**Abstract**- The semiclassical approximation and the technique of 1/n expansion are used to calculate the eigenenergies and the wave functions for the radial Schrodinger equation. It is shown that the expressions that are asymptotically exact in the limit n=n/sub r/+l+1 to infinity and which describe the above eigenenergies and the asymptotic coefficients at the origin and at infinity ensure a satisfactory precision even for states characterized by modest values of the quantum numbers n/sub r/ and l, including the ground state.

**Author**- Mur VD. Popruzhenko SV.
**Popov**VS. **Institution**- Inst. of Eng. Phys., Moscow, Russia.
**Title**- Energy and momentum spectra of photoelectrons under conditions of ionization by strong laser radiation (the case of elliptic polarization).
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.119, no.5, May 2001, pp.893-905. Publisher: MAIK Nauka, Russia.
**Abstract**- Analytical and numerical studies are made into the momentum distribution and energy spectra of photoelectrons emitted during nonlinear ionization of atoms and molecules by laser radiation with elliptic polarization. The dependence of these distributions on the ellipticity xi of an electromagnetic wave is treated, as well as their evolution upon variation of the Keldysh parameter gamma from the region of optical tunneling ( gamma [left angle bracket][left angle bracket]1) to the region of gamma [right angle bracket][right angle bracket]1, in which the ionization is multiphoton. The quasiclassical approximation is used in the calculations, in particular, the imaginary-time method and the saddle-point method with expansion in the vicinity of the field ellipse.

**Author****Popov**VS.**Institution**- State Sci. Center, Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Multiphoton atomic ionization in the field of a very short laser pulse.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.120, no.2, Aug. 2001, pp.315-32. Publisher: MAIK Nauka, Russia.
**Abstract**- Closed analytic expressions are derived for the probability of multiphoton atomic and ionic ionization in a variable electric field E(t), which are applicable for arbitrary Keldysh parameters gamma . Dependencies of the ionization probability and photoelectron pulse spectrum on the shape of a very short laser pulse are analyzed. Examples of pulse fields of various forms, including a modulated light pulse with a Gaussian or Lorentz envelope, are considered in detail. The interference effect in the photoelectron energy spectrum during atomic ionization by a periodic field of a general form is examined. The range of applicability of the adiabatic approximation in the multiphoton ionization theory is discussed. The imaginary time method is used in the calculations, which allows the probability of particle tunneling through oscillating barriers to be effectively calculated.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Schwinger mechanism of electron-positron pair production by the field of optical and X-ray lasers in vacuum.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.74, no.3, 2001, pp.151-6. Publisher: MAIK Nauka, Russia.
**Abstract**- The probability W of e/sup +/e/sup -/-pair production in vacuum by an intense time-varying electric field created by optical or X-ray laser is calculated. Two characteristic regions gamma [left angle bracket][left angle bracket]1 and gamma [right angle bracket][right angle bracket]1 of adiabaticity parameter gamma are considered. With an increase in gamma and on passing from monochromatic radiation to a finite laser pulse, the probability W increases sharply (for the same field intensity). The dependence of the probability W and the electron and positron momentum spectrum on the pulse shape is discussed (the dynamic Schwinger effect).

**Author**- Mur VD. Pozdnyakov SG.
**Popov**VS. Popruzhenko SV. **Institution**- Eng. Phys. Inst., Moscow, Russia.
**Title**- On the Zel'dovich regularization method in the theory of quasistationary states.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.75, no.5, 2002, pp.294-7. Publisher: MAIK Nauka, Russia.
**Abstract**- Complex quasienergy and level width are calculated for a weakly bound atomic state in an intense circularly polarized monochromatic laser field using the method suggested by Zel'dovich (1961) for the regularization of divergent integrals with the Gamow wave function. It is demonstrated that this method converges, and the conditions for its applicability are indicated. These results are used to discuss the accuracy of the semiclassical approximation in the problems of ionization theory.

**Author****Popov**VS.**Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- On Schwinger mechanism of e/sup +/e/sup -/ pair production from vacuum by the field of optical and X-ray lasers.
**Source**- Physics Letters A, vol.298, no.2-3, 3 June 2002, pp.83-90. Publisher: Elsevier, Netherlands.
**Abstract**- The probability W of e/sup +/e/sup -/ pair production under the action of an intense electric field created by an optical or X-ray laser is calculated. Two characteristic regions, gamma [left angle bracket][left angle bracket] 1, gamma [right angle bracket][right angle bracket] 1, of adiabaticity parameter gamma are considered. It is shown that with gamma increase, as well as when turning over from monochromatic radiation to a laser pulse of finite time duration, the probability W sharply increases (at the same field intensity). The dependence of the probability W and of the momentum spectrum of electrons and positrons on the pulse shape is discussed (the dynamical Schwinger effect).

**Author****Popov**VS.**Institution**- Inst. for Theor. & Exp. Phys., Moscow, Russia.
**Title**- The Schwinger effect and possibilities for its observation using optical and X-ray lasers.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.121, no.6, June 2002, pp.1235-48. Publisher: MAIK Nauka, Russia.
**Abstract**- The probability W of e/sup +/e/sup -/ pair production from a vacuum in an intense variable electric field generated with powerful optical or X-ray lasers is calculated. Two characteristic ranges are considered: gamma [left angle bracket][left angle bracket]1 and gamma [right angle bracket][right angle bracket]1, where gamma is the adiabaticity parameter. The probability W is shown to increase sharply with gamma (at a fixed field strength F). The dependence of W and the momentum spectrum of electrons and positrons on the laser pulse shape is discussed in detail. Numerical calculations were performed for a laser pulse with a Gaussian envelope and for some pulsed fields.

**Author**- Kazakov DI.
**Popov**VS. **Institution**- Inst. of Theor. & Exp. Phys., Acad. of Sci., Moscow, Russia.
**Title**- On the summation of divergent perturbation series in quantum mechanics and field theory.
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.122, no.4, Oct. 2002, pp.675-95. Publisher: MAIK Nauka, Russia.
**Abstract**- The possibility of recovering the Gell-Mann-Low function in the asymptotic strong-coupling regime by known first-order perturbation-theory (PT) terms beta /sub n/ and their asymptotics beta /sub n/ as n to infinity is investigated. Conditions are formulated that are necessary for recovering the required function at the physical level of rigor: (1) a large number of PT coefficients are known whose asymptotics has already been established, and (2) there is no intermediate asymptotics. Higher orders of PT, their asymptotic behavior, and power corrections are calculated in quantum mechanical problems that involve divergent PT series (including series for a funnel potential, the phi /sup 4//sub (0)/ model, and the Stark effect in a strong field). The scalar field theory phi /sup 4//sub (4)/ is considered in the (MS) and MOM regularization schemes. It is shown that one cannot make any definite conclusion about the asymptotics of the Gell-Mann-Low function as g to infinity on the basis of information available for the above theory.

**Author****Popov**VS.**Institution**- Inst. for Theor. & Exp. Phys., Moscow, Russia.
**Title**- The Zeldovich effect in a superstrong magnetic field.
**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.77, no.2, 2003, pp.79-84. Publisher: MAIK Nauka, Russia.
**Abstract**- It follows from the analysis of the precision numerical calculations of the energy spectrum of a hydrogen atom in a static magnetic field that the Zeldovich effect (rearrangement of the atomic spectrum) in the spectrum of atomic levels is observed at superstrong magnetic fields B [right angle bracket]or approximately= 5 * 10/sup 11/ G. Magnetic fields of such strengths are reached in neutron stars and magnetic white dwarfs. We established a lower bound B/sub min/ for the fields required for this effect to occur.

Information about V. S. Popov (reproduced by A. Sergeev)