Spectral concentration phenomena for the Laplace operator with the Dirichlet boundary condition on a cavity

Let D be a non-empty simply connected open bounded subset of the real space and G be its boundary of class C^2. Let (G_n) be a sequence of closed subsurfaces of G, whose union is G, which differ from G for a set of positive surface measure. The set D is the cavity associated with (G_n)_n. Let H_n,H be the self-adjoint operators on L^2 associated with minus the Laplacian on the complement of G_n,G respectively with the Dirichlet boundary condition. We show that the spectral measure associated with the operator H_n converges to the spectral measure associated with the operator H. Moreover we show that the point spectrum of H is made of an infinite number of positive eigenvalues of finite multiplicity, that the point spectrum of H_n is empty and that the essential spectrum of H_n,H is continuous and contained in the non-negative real halfline. Under the further hypothesis that G is of class C^4 and that its Gaussian curvature is positive at each point, we prove that the essential spectrum of H is continuous and coincides with the non-negative real halfline.