Method of Spectral Mappings in the Inverse Problem Theory

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.

Auteur

Vyatcheslav A. Yurko, N. G. Chernyshevsky Saratov State University, Russia.

Résumé

Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural science.

This monograph is devoted to inverse problems of spectral analysis for ordinary differential equations. Its aim ist to present the main results on inverse spectral problems using the so-called method of spectral mappings, which is one of the main tools in inverse spectral theory.

The book consists of three chapters: In Chapter 1 the method of spectral mappings is presented in the simplest version for the Sturm-Liouville operator. In Chapter 2 the inverse problem of recovering higher-order differential operators of the form, on the half-line and on a finite interval, is  considered. In Chapter 3 inverse spectral problems for differential operators with nonlinear dependence on the spectral parameter are studied.

Contenu

Inverse problems for Sturm-Liouville operators; Sturm-Liouville operators on a finite interval; uniqueness theorem; solution of the inverse problem on a finite interval; Sturm-Liouville operators on the half-line; recovery of Sturm-Liouville operators from the Weyl function; recovery of Sturm-Liouville operators from the spectral data; review of the inverse problem theory; inverse problems for higher-order differential operators; the Weyl matrix; the uniqueness theorem; solution of the inverse problem on the half-line; solution of the inverse problem on a finite interval; the self-adjoint case; differential operators with a separate spectrum; differential operators with singularities; inverse problems for pencils of differential operators; second-order differential equations with nonlinear dependence on the spectral parameter; Regge-type boundary value problems; an inverse problem for differential equations of the Orr-Sommerfeld type; inverse problems for systems of differential equations.