Spectral properties of soft quantum waveguides

TL;DR

This paper investigates the spectral properties of soft quantum waveguides modeled by Schrödinger operators, showing that sufficiently deep and narrow potential wells induce discrete bound states, with implications for quantum confinement.

Contribution

It demonstrates the existence of discrete spectrum in soft quantum waveguides with non-straight asymptotically straight curves, using the Birman-Schwinger principle, which is a novel analytical approach.

Findings

Discrete spectrum exists for sufficiently deep and narrow potential wells.

The waveguide's asymptotic straightness influences spectral properties.

Analytical methods extend understanding of quantum waveguide behavior.

Abstract

We consider a soft quantum waveguide described by a two-dimensional Schr\"odinger operators with an attractive potential in the form of a channel of a fixed profile built along an infinite smooth curve which is not straight but it is asymptotically straight in a suitable sense. Using Birman-Schwinger principle we show that the discrete spectrum of such an operator is nonempty if the potential well defining the channel profile is deep and narrow enough. Some related problems are also mentioned.