Schrödinger operators periodic in octants

Abstract

We consider Schrödinger operators with periodic potentials in the positive quadrant for dim $>1$ with Dirichlet boundary condition. We show that for any integer $N$ and any interval $I$ there exists a periodic potential such that the Schrödinger operator has $N$ eigenvalues counted with the multiplicity on this interval and there is no other spectrum on the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for other domains. The proof is based on the inverse spectral theory for Hill operators on the real line.

keywords: spectral bands, periodic Schrödinger operator, eigenvalues