Tunneling in soft waveguides:closing a book

TL;DR

This paper analyzes the spectral properties of a two-dimensional soft quantum waveguide with a 'bookcover' shape, focusing on eigenvalue accumulation, the influence of ditch profile, and the effects of potential strength.

Contribution

It provides new insights into eigenvalue behavior in soft waveguides with complex geometries, including conditions for the existence of discrete spectra.

Findings

Eigenvalues accumulate as the angle between asymptotes approaches zero.

Discrete spectrum existence depends on ditch profile and potential strength.

Numerical example demonstrates critical potential strength for spectrum emergence.

Abstract

We investigate the spectrum of a soft quantum waveguide in two dimensions of the generalized `bookcover' shape, that is, Schr\"odinger operator with the potential in the form of a ditch consisting of a finite curved part and straight asymptotes which are parallel or almost parallel pointing in the same direction. We show how the eigenvalues accumulate when the angle between the asymptotes tends to zero. In case of parallel asymptotes the existence of a discrete spectrum depends on the ditch profile. We prove that it is absent in the weak-coupling case, on the other hand, it exists provided the transverse potential is strong enough. We also present a numerical example in which the critical strength can be assessed.