Schr\"odinger operators periodic in octants

TL;DR

This paper constructs specific periodic potentials for Schr"odinger operators in multi-dimensional positive quadrants, controlling the number of eigenvalues in a given interval while maintaining essential spectrum elsewhere, using inverse spectral theory.

Contribution

It introduces a method to design periodic potentials with prescribed eigenvalue counts in a specified interval for Schr"odinger operators in higher dimensions.

Findings

Existence of potentials with exactly N eigenvalues in a given interval

Presence of essential spectrum outside the eigenvalue interval

Application of inverse spectral theory for Hill operators in multi-dimensional domains

Abstract

We consider Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger operators for other domains. The proof is based on the inverse spectral theory for Hill operators on the real line.