Norm resolvent approximation of thin homogeneous tubes by heterogeneous ones

Abstract

We study the operator [Formula: see text] restricted to a waveguide [Formula: see text], with heterogeneous function [Formula: see text] constant in the longitudinal direction. The purpose is to obtain an effective operator, in the norm resolvent sense, when the diameter of [Formula: see text] tends to zero with heterogeneity approaching a homogeneous situation (i.e. a constant function [Formula: see text]). The effective operator presents a potential that, besides the traditional dependence on waveguide geometric properties, there is also a contribution from [Formula: see text] which results, when combined with the curvature, for example, in the possibility of a repulsive interaction.