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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 2, Pages 46–53 (Mi timm1522)

Gaps in the spectrum of the Laplacian in a band with periodic delta interaction

D. I. Borisovabc

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa
b Bashkir State Pedagogical University, Ufa

Abstract: We consider the Laplace operator in an infinite planar strip with a periodic delta interaction. The width of the strip is fixed and for simplicity is chosen equal to $\pi$. The delta interaction is introduced on a periodic system of curves. Each curve consists of a finite number of segments, each having smoothness $C^1$. The curves are supposed to be strictly internal and do not intersect the boundaries of the strip. The period of their location is $2\varepsilon\pi$, where $\varepsilon$ is a sufficiently small number. The function describing the delta interaction is also periodic on the system of curves and is assumed to be bounded and measurable. The main result is the following fact. If $\varepsilon\leqslant \varepsilon_0$, where $\varepsilon_0$ is a certain explicitly calculated number and the norm of the function describing the delta interaction is smaller than some explicit constant, then a lower part of the spectrum of the operator has no internal gaps. The lower part is understood as the band of the spectrum until some point, which is explicitly calculated in terms of the parameter $\varepsilon$ as a rather simple function. This result can be considered as a first step to the proof of the strengthened Bethe-Sommerfeld conjecture on the complete absence of gaps in the spectrum of the operator for a sufficiently small period of location of delta interactions.

Keywords: periodic operator, Laplacian, delta interaction, band spectrum, absence of gaps.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00046

DOI: https://doi.org/10.21538/0134-4889-2018-24-2-46-53

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Bibliographic databases:

Document Type: Article
UDC: 517.958
MSC: 35P05, 47A10

Citation: D. I. Borisov, “Gaps in the spectrum of the Laplacian in a band with periodic delta interaction”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 2, 2018, 46–53

Citation in format AMSBIB
\Bibitem{Bor18} \by D.~I.~Borisov \paper Gaps in the spectrum of the Laplacian in a band with periodic delta interaction \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2018 \vol 24 \issue 2 \pages 46--53 \mathnet{http://mi.mathnet.ru/timm1522} \crossref{https://doi.org/10.21538/0134-4889-2018-24-2-46-53} \elib{http://elibrary.ru/item.asp?id=35060677} 

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