Inverse problems, trace formulae for discrete Schr\"odinger operators

TL;DR

This paper investigates discrete Schrödinger operators on a lattice, demonstrating potential reconstruction from the S-matrix, analyzing the spectral shift function, and estimating the discrete spectrum based on potential properties.

Contribution

It introduces a method to uniquely reconstruct potentials from the S-matrix and provides spectral estimates for trace class potentials.

Findings

Potential is uniquely reconstructed from the S-matrix at all energies.

Spectral shift function is analyzed for trace class potentials.

Discrete spectrum is estimated using moments of the spectral shift function.

Abstract

We study discrete Schroedinger operators with compactly supported potentials on the square lattice. Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function for the trace class potentials, and estimate the discrete spectrum in terms of the moments of the spectral shift function and the potential.