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Full Description
This book provides a detailed treatment of the various facets of modern Sturm-Liouville theory, including such topics as Weyl:ndash;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 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 symmetric 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.
Contents
Introduction
A bit of physical motivation
Preliminaries on ODEs
The regular problem on a compact interval $[a,b]\subset\mathbb{R}$
The singular problem on $(a,b)\subseteq \mathbb{R}$
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 $(a,b)\subseteq\mathbb{R}$
Classical oscillation theory, principal solutions, and nonprinicpal 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 $(a,b)\subseteq\mathbb{R}$ revisited
Four-coefficient Sturm-Liouville operators and distributional potential coefficients
Epilogue: Applications to some partial differnetial equations of mathematical physics
Basic facts on linear operators
Basics of spectral theory
Classes of bounded linear operators
Extensions of symmetric operators
Elements of sesquilinear forms
Basics of Nevanlinna-Herglotz functions
Bessel functions in a nutshell
Bibliography
Author index
List of symbols
Subject index
