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Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window

P. Exner·S. A. Vugalter
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Abstract

Consider the Laplacian in a straight planar strip of width $\,d\,$, with the Neumann boundary condition at a segment of length $\,2a\,$ of one of the boundaries, and Dirichlet otherwise. For small enough $\,a\,$ this operator has a single eigenvalue $\,ε(a)\,$; we show that there are positive $\,c_1,c_2\,$ such that $\,-c_1 a^4 \le ε(a)- \left(π/ d\right)^2 \le -c_2 a^4\,$. An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length $\,2a\,$ in the common boundary.

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P. Exner, S. A. Vugalter. 1996-02-02. Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window. https://arxiv.org/abs/funct-an/9602001

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