Absence of Gaps in a Lower Part of the Spectrum of a Laplacian with Frequent Alternation of Boundary Conditions in a Strip

Abstract

We consider the Laplacian in a planar infinite straight strip with frequent alternation of boundary conditions. We show that for a sufficiently small alternation period, there are no gaps in a lower part of the spectrum. In terms of certain numbers and functions, we write an explicit upper bound for the period and an expression for the length of the lower part of the spectrum in which the absence of gaps is guaranteed.