On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit

The Neumann problem in two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter. The complete asymptotic expansion for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of the matched asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order $\epsilon^2$. However, the result obtained in that way is false. The correct result can be obtained only by means of inner asymptotic expansion.