Eigenfrequencies and eigenfunctions of the Laplacian for Neumann boundary conditions in a system of two coupled cavities
We consider a region \(\Omega\) in \(\mathbb{R}^2\) that consists of two regions \(\Omega_1\) and \(\Omega_2\) with boundaries belonging to \(C^2\) and connected by a thin rectangular channel \(P\) of width \(\Omega\). Our task is to investigate the eigenvalues and eigenfunctions of the Laplacian for Neumann boundary conditions in the region \(\Omega\). We analyze a model problem in which the role of the channel is played by a line segment, and the coupling between the regions is specified by means of the technique of the theory of extensions of symmetric operators. We find the first terms in the asymptotic expansions of the eigenvalues and eigenfunctions of a model operator in the region \(\Omega\) with respect to the parameter \(\alpha\), which specifies the strength of the coupling between the regions, and we obtain an expression for the resolvent in the form of a series in powers of \(\alpha\).
asymptotic expansions of the eigenvalues
scientific article; zbMATH DE number 916202