Sturm-Liouville operators, their spectral theory, and some applications / Fritz Gesztesy, Roger Nichols, Maxim Zinchenko.

Format:

Book

Author/Creator:

Gesztesy, Fritz, 1953- author.

Nichols, Roger (Roger Allen), 1984- author.

Zinchenko, Maxim, 1980- author.

Series:

Colloquium publications (American Mathematical Society) ; v. 67.

American Mathematical Society colloquium publications, 0065-9258 ; volume 67

Language:

English

Subjects (All):

Sturm-Liouville equation.

Operator theory.

Spectral theory (Mathematics).

Physical Description:

xv, 927 pages : illustrations ; 26 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2024]

Summary:

This book provides a detailed treatment of the various facets of modern Sturm-Liouville theory, including such topics as Weyl-Titchmarsh theory, classical, renormalized, and perturbative oscillation theory, boundary data maps, traces and determinants for Sturm-Liouville operators, strongly singular Sturm-Liouville differential operators, generalized boundary values, and Sturm-Liouville operators with distributional coefficients. To illustrate the theory, the book develops an array of examples from Floquet theory to short-range scattering theory, higher-order KdV trace relations, elliptic and algebro-geometric finite gap potentials, reflectionless potentials and the Sodin-Yuditskii class, as well as a detailed collection of singular examples, such as the Bessed, generalized Bessed, and Jacobi operators. A set of appendices contains background on the basics of linear operators and spectral theory in Hilbert spaces, Schatten-von Neumann classes of compact operators, self-adjoint extensions of symmetric operators, including the Friedrichs and Krein-von Neumann entensions, boundary triplets for ODEs, Krein-type resolvent formulas, sequilinear forms, Nevanlinna-Herglotz functions, and Bessed funsctions. Provided by publisher.

Contents:

Introduction

A bit of physical motivation

Preliminaries on ODEs

The regular problem on a compact interval [a,b]⊂R

The singular problem on (a,b)⊆R

The spectral function for a problem with a regular endpoint

The 2 x 2 spectral matrix function in the presence of two singular interval endpoints for the problem on (a,b)⊆R

Classical oscillation theory, principal solutions, and nonprincipal solutions

Renormalized oscillation theory

Perturbative oscillation criteria and perturbative Hardy-type inequalities

Boundary data maps

Spectral zeta functions and computing traces and determinants for Sturm-Liouville operators

The singular problem on (a,b)⊆R revisited

Four-coefficient Sturm-Liouville operators and distributional potential coefficients

Epilogue : applications to some partial differential equations of mathematical physics

Appendix A. Basic facts on linear operators

Appendix B. Basics of spectral theory

Appendix C. Classes of bounded linear operators

Appendix D. Extensions of symmetric operators

Appendix E. Elements of sesquilinear forms

Appendix F. Basics of Nevanlinna-Herglotz functions

Appendix G. Bessel functions in a nutshell.

Notes:

Includes bibliographical references (pages 861-905) and indices.

Other Format:

Online version: Gesztesy, Fritz, 1953- Sturm-Liouville operators, their spectral theory, and some applications.

ISBN:

9781470476663

1470476665

OCLC:

1433658363