Inverse Problems for Jacobi Operators with Mixed Spectral Data

TL;DR

This paper addresses the inverse spectral problem for semi-infinite Jacobi matrices, demonstrating unique recovery from mixed spectral data and extending classical results to cases with partial spectral information.

Contribution

It introduces new methods for uniquely reconstructing Jacobi operators using combined spectral data, including cases with incomplete spectra and known norming constants.

Findings

Unique recovery of Jacobi operators from one spectrum and partial spectral data

Extension of Borg-Marchenko problem to cases with missing spectral information

Conditions under which partial spectral data suffice for reconstruction

Abstract

We consider semi-infinite Jacobi matrices with discrete spectrum. We prove that the Jacobi operator can be uniquely recovered from one spectrum and subsets of another spectrum and 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 norming constants have different index sets.