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.
Fritz Gesztesy, Baylor University, Waco, TX, Roger Nichols, The University of Tennessee at Chattanooga, TN, and Maxim Zinchenko, University of New Mexico, Albuquerque, NM.
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
