Effective quantum Hamiltonian in thin domains with non-homogeneity

We consider the Laplacian with a non-homogeneous metric in a tubular neighborhood of a compact hyper-surface in the Euclidean space of arbitrary dimension, subject to Neumann boundary conditions. It is shown that, in the limit of the width of the neighborhood shrinking to zero, the operator converges in a generalized norm-resolvent sense to an effective Laplace--Beltrami-type operator on the hyper-surface. In this way, we generalize and give an insight into the convergence of eigenvalues obtained by Yachimura [19].