Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case

Abstract:

We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in ${\mathbb {R}^2}$. This domain consists of two disjoint domains ${\Omega ^L}$, ${\Omega ^R}$ joined by a channel ${R_\varepsilon }$ of height of the order of the parameter $\varepsilon$. When an eigenvalue approaches an eigenvalue of the Laplacian in ${\Omega ^L} \cup {\Omega ^R}$, the order of convergence is $\varepsilon$, while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: $\varepsilon \left | {{\text {ln}}\varepsilon } \right |$. We also obtain estimates on the behavior of the eigenfunctions.