Spectral Theory and Wave Processes de M. Sh. Birman

Spectral Theory and Wave Processes: Topics in Mathematical Physics, cartea 1

M. Sh. Birman

The articles in this collection are devoted to various problems in mathematical physics and mathematical analysis, primarily in the fields of spectral theory and the theory of wave processes. The collection is intended for mathematicians sPf>cializing in the fields of mathematical physics, functional analysis, and the theory of differential equations. In addition, it is of some interest to theoretical physicists. The first paper deals with a mixed boundary value problem for a system of elasticity equations, and considers fields in the neighborhood of various wave fronts. The method used permits an estimate of the errors in the Ben-Menahem approximate method. Paper 2 investigates operators in separable Hilbert space given by double integrals of a type defined at the beginning of the paper, and in which integration is understood as the limit of the integral sums of Riemann-stieltjes. In paper 3, the problem of calculation of elastic constants for a laminarly inhomogeneous semi-infinite medium is conSidered, and the uniqueness of the solution of the inverse seismic problem at finite depth proved. The fourth paper gives a detailed account of the results of an earlier paper by the same author, in which he generalized to the three-dimensional case the trace formulas obtained for the one-dimensional Schroedinger operator. Asymptotic estimates of the resolvent kernel and solutions of the scattering problem are given.

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Cuprins

The Lamb Problem for an Inhomogeneous Elastic Half-Space.- § 1. Formulation of the Problem.- § 2. Exact Solution.- § 3. Asymptotic Behavior of the Solutions of a System.- § 4. Geometry of Rays and Wave Fronts.- § 5. The Field in the Neighborhood of the Ordinary Fronts of the Longitudinal and Transverse Waves.- § 6. The Field in the Neighborhood of the Rayleigh Wave Front.- § 7. The Field in the Neighborhood of the Slip Fronts of the Longitudinal and Transverse Waves.- § 8. The Field in the Neighborhood of the Front of the Head Wave.- § 9. Formulas for the Axially Symmetric Lamb Problem.- § 10. Conclusions.- Literature Cited.- Stieltjes Double-Integral Operators.- § 1. Introduction.- § 2. Transformers of Class (S2, S2).- § 3. Transformers of Class (S1, Sq).- § 4. Subsidiary Information About Stieltjes Integrals.- § 5. General Theorems on Transformers.- § 6. Some Remarks About Special Transformers.- Literature Cited.- The Inverse Problem in the Theory of Seismic Wave Propagation.- § 1. The Inverse Problem for the String Equation.- § 2. The Inverse Seismic Problem.- Literature Cited.- The Trace Formulas and Some Asymptotic Estimates of the Resolvent Kernel of the Three-Dimensional Schroedinger Equation.- § 1. Asymptotic Estimates.- § 2. Calculations of Sp(R? — R?0).- § 3. Trace Formulas.- Literature Cited.- The Nonself-Adjoint Schroedinger Operator.- § 1. The One-Dimensional Schroedinger Operator.- § 2. Expansion Theorem.- § 3. The Three-Dimensional Schroedinger Operator.- Literature Cited.