Sturm-Liouville operators, their spectral theory, and some applications

By: Gesztesy, FritzContributor(s): Nichols, Roger | Zinchenko, MaximMaterial type: TextTextLanguage: English Series: Colloquium publications ; 67Publication details: American Mathematical Society Rhode Island 2024Description: xv, 928pISBN: 9781470476663 (HB)Subject(s): Sturm-Liouville equation | Operator theory | Spectral theory (Mathematics) | Ordinary differential equations -- Boundary value problems -- Sturm-Liouville theory | Ordinary differential equations -- Ordinary differential operators -- Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions | Operator theory -- Special classes of linear operators -- Symmetric and selfadjoint operators (unbounded) | Mathematics
Contents:
Preface 1. Introduction 2. A Bit of Physical Motivation 3. Preliminaries on ODEs 4. The Regular Problem on a Compact Interval [a,b] R 5. The Singular Problem on (a,b) R 6. The Spectral Function for a Problem with a Regular Endpoint 7. The 2 2 Spectral Matrix Function in the Presence of Two Singular Interval Endpoints for the Problem on (a,b) R 8. Classical Oscillation Theory, Principal Solutions, and Nonprincipal Solutions 9. Renormalized Oscillation Theory 10. Perturbative Oscillation Criteria and Perturbative Hardy-Type Inequalities 11. Boundary Data Maps 12. Spectral Zeta Functions and Computing Traces and Determinants for Sturm-Liouville Operators 13. The Singular Problem on (a,), R Revisited 14. Four-Coefficient Sturm-Liouville Operators and Distributional Potential Coefficients 15. Epilogue: Applications to Some Partial Differential Equations of Mathematical Physics
Summary: his 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 Strum-Liouville differential operators, generalized boundary values, and Strum-Liouville operators with distributional coefficients. To illustrate the teory, the book develops an array of examples from Floquet theory to short-range scattering theory, higher order KdV trace relations, elloptic 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 Bessel, generalized Bessel, 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 summetric operators, including the Friedrichs and Krein-von Neumann extensions, boundary triplets for ODEs, Krein-type resolvent formulas, sesquilinear forms, Nevanlinna-Herglotz functions, and Bessel functions.
Item type: BOOKS
Tags from this library: No tags from this library for this title. Log in to add tags.
    Average rating: 0.0 (0 votes)

Includes index.

Includes bibliography (p. 861-905)

Preface
1. Introduction
2. A Bit of Physical Motivation
3. Preliminaries on ODEs
4. The Regular Problem on a Compact Interval [a,b] R
5. The Singular Problem on (a,b) R
6. The Spectral Function for a Problem with a Regular Endpoint
7. The 2 2 Spectral Matrix Function in the Presence of Two Singular Interval Endpoints for the Problem on (a,b) R
8. Classical Oscillation Theory, Principal Solutions, and Nonprincipal Solutions
9. Renormalized Oscillation Theory
10. Perturbative Oscillation Criteria and Perturbative Hardy-Type Inequalities
11. Boundary Data Maps
12. Spectral Zeta Functions and Computing Traces and Determinants for Sturm-Liouville Operators
13. The Singular Problem on (a,), R Revisited
14. Four-Coefficient Sturm-Liouville Operators and Distributional Potential Coefficients
15. Epilogue: Applications to Some Partial Differential Equations of Mathematical Physics

his 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 Strum-Liouville differential operators, generalized boundary values, and Strum-Liouville operators with distributional coefficients. To illustrate the teory, the book develops an array of examples from Floquet theory to short-range scattering theory, higher order KdV trace relations, elloptic 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 Bessel, generalized Bessel, 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 summetric operators, including the Friedrichs and Krein-von Neumann extensions, boundary triplets for ODEs, Krein-type resolvent formulas, sesquilinear forms, Nevanlinna-Herglotz functions, and Bessel functions.

There are no comments on this title.

to post a comment.
The Institute of Mathematical Sciences, Chennai, India

Powered by Koha