Bound states in weakly deformed strips and layers

TL;DR

This paper investigates the existence and asymptotic behavior of bound states in weakly deformed straight strips and layers, extending previous results to three dimensions and providing criteria for bound state formation.

Contribution

It generalizes a known result to 3D structures, derives the leading order of weak-coupling asymptotics, and introduces an alternative method to evaluate higher-order terms.

Findings

Bound states exist if the deformation increases volume.

Derived the leading order of weak-coupling asymptotics.

Established a criterion for bound states when volume change is zero.

Abstract

We consider Dirichlet Laplacians on straight strips in R^2 or layers in R^3 with a weak local deformation. First we generalize a result of Bulla et al. to the three-dimensional situation showing that weakly coupled bound states exist if the volume change induced by the deformation is positive; we also derive the leading order of the weak-coupling asymptotics. With the knowledge of the eigenvalue analytic properties, we demonstrate then an alternative method which makes it possible to evaluate the next term in the asymptotic expansion for both the strips and layers. It gives, in particular, a criterion for the bound-state existence in the critical case when the added volume is zero.