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.

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

References in corpus (4)

• Convergence of spectra of graph-like thin manifolds
• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Opening up and control of spectral gaps of the Laplacian in periodic domains

Cited by in corpus (1)

• Trace Hardy inequality for the Euclidean space with a cut and its applications