Lower bounds on the interaction between cavities connected by a thin tube
The authors are concerned with a spectrum of the Dirichlet Laplacian in a Dumbbell shaped domain, which consists of 2 disjoint regions and a thin channel connecting them. It is interesting that the feature of extremal shape of the domain arises in the analytic quantity. The authors give a lower bound of the difference of the first 2 eigenvalues \(E_ 1\), \(E_ 2\) in terms of the size of the section of the thin channel. Such a study (splitting of eigenvalues) is an important subject in the spectral problem in many areas including the quantum mechanics (tunnel effect).
scientific article; zbMATH DE number 572055
scientific article