Soft quantum waveguides in three dimensions

TL;DR

This paper investigates the spectral properties of a three-dimensional soft quantum waveguide with an infinite tube, demonstrating that certain geometric conditions preserve the essential spectrum and can induce discrete bound states.

Contribution

It introduces conditions under which the essential spectrum remains unchanged and provides criteria for the existence of discrete spectrum in 3D soft quantum waveguides.

Findings

Essential spectrum unaffected by smooth bends under certain conditions

Derived geometric criteria for nonempty discrete spectrum

Established the influence of tube geometry on spectral properties

Abstract

We discuss a three-dimensional soft quantum waveguide, in other words, Schr\"odinger operator in $\R^3$ with an attractive potential supported by an infinite tube and keeping its transverse profile fixed. We show that if the tube is asymptotically straight, the distance between its ends is unbounded, and its twist satisfies the so-called Tang condition, the esential spectrum is not affected by smooth bends. Furthermore, we derive a sufficient condition, expressed in terms of the tube geometry, for the discrete spectrum of such an operator to be nonempty.