http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Sturm-Liouville operators, their spectral theory, and some applications
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.
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.
Fritz Gesztesy, Roger Nichols, Maxim Zinchenko.
Includes bibliographical references and indexes.
Colloquium publications (American Mathematical Society) ; v. 67