Sturm-Liouville Operators, Their Spectral Theory, and Some Applications

Fritz Gesztesy (Author), Roger Nichols (Author), Maxim Zinchenko (Author)

9781470476663

Hardback, published 31 October 2024

927 pages
25.4 x 17.8 x 5.5 cm, 1.846 kg

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.

• 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