Journal of Mathematical Physics, Analysis, Geometry
2018, vol. 14, No 3, pp. 270-285   https://doi.org/10.15407/mag14.03.270     ( to contents , go back )
https://doi.org/10.15407/mag14.03.270

Gap Control by Singular Schrödinger Operators in a Periodically Structured Metamaterial

Pavel Exner

Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavní 130, Řež near Prague, 25068, Czech Republic

Doppler Institute, Czech Technical University, Břehová 7, Prague, 11519, Czech Republic
E-mail: exner@ujf.cas.cz

Andrii Khrabustovskyi

Institute of Applied Mathematics, Graz Institute of Technology, Steyrergasse 30, Graz, 8010, Austria
E-mail: khrabustovskyi@math.tugraz.at

Received February 21, 2018.

Abstract

We consider a family $\{\mathcal{H}_\varepsilon\}_{\varepsilon}$ of $\varepsilon\mathbb{Z}^n$-periodic Schrödinger operators with $\delta'$-interactions supported on a lattice of closed compact surfaces; within a minimum 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.

Mathematics Subject Classification 2000: 35P05, 35P20, 35J10, 35B27
Key words: periodic Schrödinger operators, δ' interaction, spectral gaps, eigenvalue asymptotics.

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