An algorithm for finding the periodic potential of the three-dimensional Schrödinger operator from spectral invariants

Abstract

In this paper, we investigate the three-dimensional Schrödinger operator with a periodic, relative to a lattice Ω of 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 ∊ V of the three-dimensional Schrödinger 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 ∊ V.