Gap control by singular Schrödinger operators in a periodically structured metamaterial

Abstract

We consider a family $\{{\mathcal{H}}^{\epsilon }{\}}_{\epsilon >0}$ of $\epsilon {\mathbb{Z}}^n$-periodic Schrödinger operators with ${\delta }^{\prime }$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in \mathbb{N}$ surfaces. We show that in the limit when $\epsilon \to 0$ and the interactions strengths are appropriately scaled, ${\mathcal{H}}^{\epsilon }$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

This paper is dedicated to Volodymyr Olexandrovych Marchenko on the occasion of his jubilee