Effective quantum Hamiltonian in thin domains with non-homogeneity

TL;DR

This paper studies the behavior of the Laplacian with a non-homogeneous metric in thin tubular domains around hypersurfaces, showing it converges to an effective operator on the hypersurface as the domain shrinks.

Contribution

It generalizes previous results by establishing convergence of the Laplacian with non-homogeneous metrics to an effective operator in arbitrary dimensions.

Findings

Convergence of the Laplacian to an effective operator as the domain width shrinks.

Extension of Yachimura's eigenvalue convergence results to more general settings.

Insight into spectral properties of Laplacians in thin, non-homogeneous domains.

Abstract

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).