  We consider the Laplacian with a non-homogeneous metric in a tubular
neighbourhood of a compact hypersurface in the Euclidean space of arbitrary
dimension, subject to Neumann boundary conditions. It is shown that, in the
limit of the width of the neighbourhood shrinking to zero, the operator
converges in a generalised norm-resolvent sense to an effective
Laplace-Beltrami-type operator on the hypersurface. In this way, we generalise
and give an insight into the convergence of eigenvalues obtained by Yachimura
(arXiv:1706.05027).
