Complex Analytic Approaches to Inverse Spectral Problems

Abstract

In this dissertation we consider methods from complex analysis to solve inverse spectral problems for Schroedinger operators on finite intervals and semi-infinite Jacobi operators. After discussing necessary background from complex function theory and harmonic analysis in Chapter 2, we consider Schroedinger operators on a finite interval with an L1-potential in Chapter 3. We prove that the potential can be uniquely recovered from one spectrum and subsets of another spectrum and point masses of the spectral measure (or norming constants) corresponding to the first spectrum. We also solve this Borg-Marchenko-type problem under some conditions on two spectra, when missing part of the second spectrum and known point masses of the spectral measure have different index sets. In Chapter 4, we consider semi-infinite Jacobi matrices with discrete spectrum. We prove that a Jacobi operator can be uniquely recovered from one spectrum and subsets of another spectrum and norming constants corresponding to the first spectrum. As a corollary, we obtain semi-infinite Jacobi analog of Marchenko’s inverse spectral theorem for Schroedinger operators, i.e. a Jacobi operator can be uniquely recovered from the Weyl m-function (or the spectral measure). We also solve our Borg-Marchenko-type problem under some conditions on two spectra, when missing part of the second spectrum and known norming constants have different index sets.