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 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
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Contents
Volume 7 | 2 |
3 Formulation of the main problems | 11 |
5 The case 21 d Spectrum estimates to the left of 2 1 | 27 |
7 A noninterpolation approach | 33 |
8 Examples of explicit asymptotic formulas | 42 |
9 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
A₁ A₁(aV apply arbitrary assume assumptions asymptotic formula b₁ bijection Birman bounded operator boundedness coincides compact operators consider the operator Corollary corresponding defined denote discrete spectrum distribution function eigenvalues elliptic operator English transl equation 1.1 equivalent estimates example finite Fourier function spaces given H₂ Hardy inequality Hilbert space holds implies integral operator interpolation inverse operator kernel L₁ L₂ L₂(p L₂(R Lemma lim sup linear loc(R Math metric norm notation number of eigenvalues obtain operator A¹ operator G perturbations potential projection operator PROOF properties Proposition prove quadratic form quasinorm quotient relation s-numbers Schrödinger operator selfadjoint selfadjoint operator semibounded singular numbers Sobolev Sobolev space solution statement subsection subspace sufficient Suppose symbol Theorem 1.2 total reflection operator V\u² dx λ₂