**Advances in Soviet Mathematics**

1991;
204 pp;
Hardcover

MSC: Primary 34; 35; 45; 47;
**Print ISBN: 978-0-8218-4106-8
Product Code: ADVSOV/7**

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# Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations

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*M. Sh. Birman*

The Leningrad Seminar on mathematical physics, begun in 1947 by V. I.
Smirnov and now run by O. A. Ladyzhenskaya, is sponsored by Leningrad
University and the Leningrad Branch of the Steklov Mathematical Institute of
the Academy of Sciences of the USSR. The main topics of the seminar center on
the theory of boundary value problems and related questions of analysis and
mathematical physics. This volume contains adaptations of lectures presented
at the seminar during the academic year 1989-1990.

For the most part, the papers are devoted to investigations of the spectrum
of the Schrödinger operator (or its generalizations) perturbed by some
relatively compact operator. The book studies the discrete spectrum that
emerges in the spectral gaps of the nonperturbed operator, and considers the
corresponding estimates and asymptotic formulas for spectrum distribution
functions in the large-coupling-constant limit. The starting point here is the
opening paper, which is devoted to the important case of a semi-infinite gap.
The book also covers the case of inner gaps, related questions in the theory of
functions, and an integral equation with difference kernel on a finite
interval. The collection concludes with a paper focusing on the classical
problem of constructing scattering theory for the Schrödinger operator
with potential decreasing faster than the Coulomb potential.