Bound-state asymptotic estimates for window-coupled Dirichlet strips and layers

TL;DR

This paper investigates the discrete spectrum of the Dirichlet Laplacian on coupled strips and layers, providing asymptotic estimates for the eigenvalue gap and proposing a conjecture on weak-coupling behavior.

Contribution

It offers new asymptotic bounds for eigenvalues in coupled geometric structures and extends understanding of weak-coupling limits in these systems.

Findings

Derived upper and lower bounds on eigenvalue gaps

Established asymptotic estimates for small window sizes

Formulated a conjecture on weak-coupling asymptotic behavior

Abstract

We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel 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.