Inverse problems for Jacobi operators with mixed spectral data

Abstract

ABSTRACT 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 Schödinger 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.