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

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 $\epsilon (a)$; we show that there are positive $c_1,c_2$ such that $-c_1{a}^4\le \epsilon (a)-{\left(\pi /d\right)}^2\le -c_2{a}^4$. An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length $2a$ in the common boundary.

LaTeX file, 12 pages, no figures

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