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Zh. Mat. Fiz. Anal. Geom., 2018, Volume 14, Number 3, Pages 270–285 (Mi jmag700)  

Gap control by singular Schrödinger operators in a periodically structured metamaterial

Pavel Exnerab, Andrii Khrabustovskyic

a Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavní 130, Řežnear Prague, 25068, Czech Republic
b Doppler Institute, Czech Technical University, Břehová 7, Prague, 11519, Czech Republic
c Institute of Applied Mathematics, Graz Institute of Technology, Steyrergasse 30, Graz, 8010, Austria

Abstract: We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ 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.

Key words and phrases: periodic Schrödinger operators, $\delta'$ interaction, spectral gaps, eigenvalue asymptotics.

Funding Agency Grant Number
Grantová Agentura České Republiky 17-01706S
Czech–Austrian Grant CZ 02/2017 7AMBL7ATO22
Austrian Science Fund M 2310-N32
The research was supported by Project No. 17-01706S of the Czech Science Foundation (GAČR) and by the Czech–Austrian Grant CZ 02/2017 7AMBL7ATO22. A.K. is supported by the Austrian Science Fund (FWF) under Project No. M 2310-N32.


DOI: https://doi.org/10.15407/mag14.03.270

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Bibliographic databases:

MSC: 35P05, 35P20, 35J10, 35B27
Received: 21.02.2018
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Citation: Pavel Exner, Andrii Khrabustovskyi, “Gap control by singular Schrödinger operators in a periodically structured metamaterial”, Zh. Mat. Fiz. Anal. Geom., 14:3 (2018), 270–285

Citation in format AMSBIB
\Bibitem{ExnKhr18}
\by Pavel~Exner, Andrii~Khrabustovskyi
\paper Gap control by singular Schr\"odinger operators in a periodically structured metamaterial
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2018
\vol 14
\issue 3
\pages 270--285
\mathnet{http://mi.mathnet.ru/jmag700}
\crossref{https://doi.org/10.15407/mag14.03.270}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000450683100002}


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