Direct and inverse spectral theory of singular left-definite Sturm-Liouville operators

Abstract

We discuss direct and inverse spectral theory of self-adjoint Sturm-Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl-Titchmarsh theory for these relations. Consequently, we apply de Branges' subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa-Holm equation.

30 pages, see extended version v2 for more detailed proofs

References in corpus (2)

• Weyl-Titchmarsh Theory for Schroedinger Operators with Strongly Singular Potentials
• On the isospectral problem of the dispersionless Camassa-Holm equation

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• The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation
• Singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators
• On the absolutely continuous spectrum of generalized indefinite strings
• On Spectral Deformations and Singular Weyl Functions for One-Dimensional Dirac Operators
• Unique solvability of a coupling problem for entire functions
• Two inverse spectral problems for a class of singular Krein strings
• Spectral Properties of Singular Sturm-Liouville Operators via Boundary Triples and Perturbation Theory
• Trace formulas and inverse spectral theory for generalized indefinite strings
• On the absolutely continuous spectrum of generalized indefinite strings II
• Generalized indefinite strings with purely discrete spectrum
• The conservative Camassa-Holm flow with step-like irregular initial data