Inverse spectral problem for the Schr\"odinger operator on the square lattice

TL;DR

This paper addresses the inverse spectral problem for Schrödinger operators on a square lattice quantum graph, demonstrating unique potential recovery from boundary data and the S-matrix, with a reduction to discrete operators.

Contribution

It introduces a novel reconstruction method for potentials on a square lattice quantum graph using boundary maps and S-matrix data, extending inverse spectral theory.

Findings

Dirichlet-to-Neumann map uniquely determines potentials

Reconstruction reduces to a discrete Schrödinger operator

S-matrix over an energy set uniquely specifies potentials

Abstract

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schr\"odinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.