Estimates and Asymptotics for Discrete Spectra of Integral and Differential EquationsThe 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 19891990. 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 largecouplingconstant limit. The starting point here is the opening paper, which is devoted to the important case of a semiinfinite 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

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Contents
Volume 7  2 
SH BIRMAN AND M Z SOLOMYAK  8 
fundamental estimates and interpolation  14 
5 The case 21 d Spectrum estimates to the left of a  27 
A noninterpolation approach  33 
8 Examples of explicit asymptotic formulas  42 
Magnetic Schrödinger operator Periodic operator  49 
Discrete Spectrum in the Gaps of a Continuous One for Perturbations  54 
Mathematics Subject Classification Primary 35J10 35P20  57 
Discrete Spectrum in the Gaps for Perturbations of the Magnetic  73 
Reflection Operators and Their Applications to Asymptotic  107 
Weyl Asymptotics for the Discrete Spectrum of the Perturbed Hill  159 
On Solutions of the Schrödinger Equation with Radiation Conditions  179 
Common terms and phrases
According allows apply approach arbitrary assertion assume assumptions asymptotic formula bijection bounded closed coincides compact condition consequence consider constant constructed contains continuous Corollary corresponding defined definition denote depend derived differential discrete spectrum discuss distribution eigenvalues English transl equality equation equivalent estimates evident example exists fact factor finite fixed function give given holds implies inequality integral interpolation interval introduce inverse operator leads Lemma limit linear mapping Math means measure Moreover norm notation Note obtain particular periodic perturbations possible potential present problem proof properties Proposition prove reflection operator relation REMARK replaced representation respect restrict satisfies Schrödinger operator selfadjoint similar singular solution space spectral statement subsection subspace sufficient Suppose symbol taking into account Theorem theory transformation true turns verify