Pavel Exner and Andrii Khrabustovskyi
Gap control by singular Schr\"odinger operators in a periodically structured metamaterial
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ABSTRACT.  We consider a family $\{\mathcal{H}_arepsilon\}_{arepsilon}$ of $arepsilon\mathbb{Z}^n$-periodic Schr\"odinger operators with $\delta'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\N$ surfaces. We show that in the limit when $arepsilon	o 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}_arepsilon$ 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.