Exponential splitting of bound states in a waveguide with a pair of distant windows

Abstract

We consider Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann ``windows'' of the same length the centers of which are $2l$ apart, and study the asymptotic behaviour of the discrete spectrum as $l\to \infty$. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit $l\to \infty$. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue.

References in corpus (2)

• Bound states and scattering in quantum waveguides coupled laterally through a boundary window
• Geometric coupling thresholds in a two-dimensional strip

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