Mixed data in inverse spectral problems for the Schrödinger operators

Abstract

We consider the Schrödinger operator on a finite interval with an L 1 -potential. 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.

Uniqueness theorems for meromorphic inner functions and canonical systems

“…The interlacing property of {a n } and {b n } implies convergence of the infinite products in (2.2) and (2.3). Then Lemma 4.1 in [5] proves these representations for the m-function of the Schrödinger operator on (0, 1) with Dirichlet-Dirichlet boundary conditions. The m-function is a meromorphic Herglotz function, its sets of poles and zeros are bounded below and satisfy some asymptotic properties.…”

Inverse Problems for Jacobi Operators with Mixed Spectral Data

“…One can find the statements of these classical theorems and some other results from the inverse spectral theory of Schrödinger operators e.g. in [16] and references therein.…”

Ambarzumian-type Problems for Discrete Schrödinger Operators

“…The study of inverse spectral problems of Schrödinger (Sturm-Liouville) equations goes back to Ambarzumian's work on a finite interval [1], and there are vast and still expanding literature on both continuous (see e.g. [8,12,13,15,20] and references therein) and discrete (see e.g. [2][3][4][5][6][7][8][9][10][11]16,18] and references therein) settings.…”