RATES OF EIGENVALUES ON A DUMBBELL DOMAIN - SIMPLE EIGENVALUE CASE

We obtain the first term in the asymptotic expansion of the eigenvalue s of the Laplace operator in a typical dumbbell domain in R(2). This d omain 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. Wh en an eigenvalue approaches an eigenvalue of the Laplacian in Omega(L) boolean OR Omega(R), the order of convergence is epsilon, while if th e eigenvalue approaches an eigenvalue which comes from the channel, th e order is weaker:epsilon\In epsilon\. We also obtain estimates on the behavior of the eigenfunctions.