Soft quantum waveguides with an explicit cut-locus

TL;DR

This paper investigates the spectral properties of 2D Schrödinger operators with specific attractive potentials along a curved channel, establishing the existence of discrete eigenvalues using a novel geometric approach.

Contribution

It introduces a new technique to prove eigenvalue existence in non-smooth waveguides without relying on non-positive constraining potentials.

Findings

Discrete eigenvalues are proven to exist for the considered waveguide model.

The method applies parallel coordinates outside the cut-locus of the curve.

The approach extends previous results to non-smooth geometries.

Abstract

We consider two-dimensional Schroedinger operators with an attractive potential in the form of a channel of a fixed profile built along an unbounded curve composed of a circular arc and two straight semi-lines. Using a test-function argument with help of parallel coordinates outside the cut-locus of the curve, we establish the existence of discrete eigenvalues. This is a special variant of a recent result of Exner in a non-smooth case and via a different technique which does not require non-positive constraining potentials.