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We consider the Laplacian with a non-homogeneous metric in a tubularneighbourhood of a compact hypersurface in the Euclidean space of arbitrarydimension, subject to Neumann boundary conditions. It is shown that, in thelimit of the width of the neighbourhood shrinking to zero, the operatorconverges in a generalised norm-resolvent sense to an effectiveLaplace-Beltrami-type operator on the hypersurface. In this way, we generaliseand give an insight into the convergence of eigenvalues obtained by Yachimura(arXiv:1706.05027).

Effective quantum Hamiltonian in thin domains with non-homogeneity