Topologically non-trivial quantum layers

TL;DR

This paper investigates the spectral properties of quantum layers built over complex, non-polar surfaces in three-dimensional space, extending previous results to more topologically intricate geometries.

Contribution

It generalizes spectral analysis of quantum layers to surfaces with handles and multiple ends, using an intrinsic geometric approach.

Findings

Extended spectral results to surfaces without poles

Analyzed layers over surfaces with handles and multiple ends

Discussed spectral properties of deformed layers

Abstract

Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain layers.