An Algorithm for Finding the Periodic Potential of the Three-dimensional Schrodinger Operator from the Spectral Invariants

TL;DR

This paper presents an algorithm to uniquely determine a class of three-dimensional periodic potentials in the Schrödinger operator from spectral invariants, and analyzes its stability and uniqueness.

Contribution

The paper introduces a constructive algorithm for identifying a specific class of periodic potentials from spectral data in three dimensions.

Findings

The algorithm can uniquely determine the potential from spectral invariants.

The stability of the algorithm with respect to spectral data is analyzed.

Uniqueness of the potential within a large class is established.

Abstract

In this paper, we investigate the three-dimensional Schrodinger operator with a periodic, relative to a lattice {\Omega} of R3, potential q. A special class V of the periodic potentials is constructed, which is easily and constructively determined from the spectral invariants. First, we give an algorithm for the unique determination of the potential q in V of the three-dimensional Schrodinger operator from the spectral invariants that were determined constructively from the given Bloch eigenvalues. Then we consider the stability of the algorithm with respect to the spectral invariants and Bloch eigenvalues. Finally, we prove that there are no other periodic potentials in the set of large class of functions whose Bloch eigenvalues coincides with the Bloch eigenvalues of q in V.