Unsuccessful application for a grant of American Mathematical Society

                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

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        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.