Geodesic
On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex Analysis. Mathematical Physics, Tome 162 (2019), pp. 3-14

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In this paper, we consider the Laplace operator in a flat strip whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
Keywords: Bethe–Sommerfeld hypothesis, Laplacian, strip, oscillating boundary. 
@article{INTO_2019_162_a0,
     author = {D. I. Borisov},
     title = {On lacunas in the spectrum of the {Laplacian} with the {Dirichlet} boundary condition in a strip with oscillating boundary},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--14},
     publisher = {mathdoc},
     volume = {162},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_162_a0/}
}
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D. I. Borisov. On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex Analysis. Mathematical Physics, Tome 162 (2019), pp. 3-14. http://geodesic.mathdoc.fr/item/INTO_2019_162_a0/