From A.V.Sergeev S.I.Vavilov State Optical Institute Tuchkov per. 1, St.-Petersburg, 199034 Russia To Tim Goggins American Mathematical Society P.O.Box 6248 Providence, R.I. 02940-6248 USA --------------------------------------------------------------------------- APPLICATION FOR A GRANT OF AMERICAN MATHEMATICAL SOCIETY Sergeev Aleksei Vital'evich, May 18, 1957. Address: flat 14, pr.Raevskogo 24, St.-Petersburg, 194064 USSR. Phone number: 552-67-67. Place of work: S.I.Vavilov State Optical Institute, Tuchkov per. 1, St.-Petersburg, 199034 USSR, e-mail: SERGEEV@soi.spb.su Publications 1. Sergeev A.V., Sherstyuk A.I. Higher orders and structure of perturbation-theory series for screened Coulomb potential. JETP 1982, v.55, no.4, p.625. 2. Sergeev A.V., Sherstyuk A.I. High-order perturbation-theory for the bound states of the Dirac-equation with a Yukawa- type potential. Sov.J. of Nucl.Phys 1984, v.39, no.5, p.731. 3. Sergeev A.V. Recursive algorithm for Pade - Hermite approximants. Zhournal vychislitel'noi matematiki i matematicheskoi fiziki 1986, v.26, no.3, p.348. 4. Vainberg V.M., Mur V.D., Popov V.S., Sergeev A.V. Strong- field Stark effect. JETP Lett. 1986, v.44, no.1, p.9.; Stark effect for the Rydberg states of the hydrogen atom. JETP Lett. 1987, v.46, no.5, p.225; The hydrogen atom in a strong electric field. Sov.Phys.-JETP 1987, v.66, no.2, p.258. 5. Vainberg V.M., Mur V.D., Popov V.S., Sergeev A.V., Scheblykin A.V. 1/n-expansion in quantum mechanics. Teor. i Mat.Fiz. 1988, v.74, no.3, p.399; Stark resonances and scaling in Rydberg atoms. ZhETF 1989, v.96, no.1, p.91 (there is English translation in Sov.Phys.-JETP). 6. Sergeev A.V. 1/N-expansion for the three-body problem. Yadernaya fizika 1989, v.50, no.4, p.945 (there is English translation in Sov.J. of Nucl.Phys.). 7. Mur V.D., Popov V.S., Sergeev A.V. 1/N-expansion in quantum mechanics. Sov.Phys.-JETP 1990, v.70, no.1, p.16. 8. V.S.Popov, V.D.Mur, A.V.Sergeev, V.M.Weinberg. Strong-field Stark effect: perturbation theory and 1/n-expansion. Phys. Lett.A 1990, v.149, no.9, p.418; 1/n-expansion and scaling for the Stark effect in Rydberg atoms. Phys. Lett.A 1990, v.149, no.9, p.425. 9. Popov V.S., Mur V.D., Sergeev A.V. Quantization rules with barrier penetrability included. J.Moscow Phys.Soc. 1991, v.1, no.1, p.15.; Generalization of Gamov formala on multidimensional case. Pis'ma v ZhETF 1991, v,53, no.9, p.433; Yadernaya Fizika 1991, v.54, no.4, p.950; Quantization rules including barrier penetrability. ZhETP 1991, v.100, no.1, p.20 (there are English translations). 10. Sergeev A.V. Positron-nucleus resonances in electric and magnetic fields. Yadernaya Fizika (Sov.J. of Nucl.Phys.) 1991, v.54, no.5, p.1225. 11. Popov V.S., Sergeev A.V., Shcheblykin A.V. Structure of high orders of 1/n-expansion. ZhETF, 1992, v.102, no.11, p.1453. Summary of research activity The work is directed on the development of perturbation theory (PT) methods in quantum mechanics in order to make it possible to apply them in the region where the perturbation parameter cease to be small . With this purpose a large number of the terms of PT series is calculated, then such series are summed. On the first stage the problem is reduced to the determination of high order PT terms (derivation of the reccurence relations) and to the calculation on computer sufficiently large number of PT coefficients. Usually the PT series appears to be divergent. On the secound stage, the suitable method for summation of the divergent series (such as Pade or Borel summation procedure) is chosen, the physically significant values are calculated and the results are compared with the values obtained by independent methods. Now I am making extensive computer experiments with asymptotic expansions, its large-order asymptotics, and summation for various quantum- mechanical problems. The name of the Russian institution that will administer the grant: Sankt-Petersburg Mathematical Society or Leningrad State University.

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