Exner P., Vugalter S.A.
Bound-state asymptotic estimates for window--coupled
Dirichlet strips and layers
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ABSTRACT. We consider the discrete spectrum of the Dirichlet Laplacian on a
manifold consisting of two adjacent parallel straight strips or
planar layers coupled by a finite number $N$ of windows in the
common boundary. If the windows are small enough, there is just one
isolated eigenvalue. We find upper and lower asymptotic bounds on the
gap between the eigenvalue and the essential spectrum in the planar
case, as well as for $N=1$ in three dimensions. Based on these
results, we formulate a conjecture on the weak-coupling asymptotic
behaviour of such bound states.