RATES OF EIGENVALUES ON A DUMBBELL DOMAIN - SIMPLE EIGENVALUE CASE

We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in R(2). This domain consists of two disjoint domains Omega(L), Omega(R) joined by a channel R(e)psilon of height of the order of the parameter epsilon. When an eigenvalue approaches an eigenvalue of the Laplacian in Omega(L) boolean OR Omega(R), the order of convergence is epsilon, while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker:epsilon\In epsilon\. We also obtain estimates on the behavior of the eigenfunctions.