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We consider the Laplace operator with Dirichlet boundary conditions on aplanar domain and study the effect that performing a scaling in one directionhas on the spectrum. We derive the asymptotic expansion for the eigenvalues andcorresponding eigenfunctions as a function of the scaling parameter aroundzero. This method allows us, for instance, to obtain an approximation for thefirst Dirichlet eigenvalue for a large class of planar domains, under very mildassumptions.

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains