Gap control by singular Schr\"odinger operators in a periodically structured metamaterial

Preprint · February 2018with 17 Reads 
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Abstract
We consider a family $\{\mathcal{H}_\varepsilon\}_{\varepsilon}$ of $\varepsilon\mathbb{Z}^n$-periodic Schr\"odinger operators with $\delta'$-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 $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}_\varepsilon$ 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.

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