Conformal Mapping and Bound States in Bent Waveguides

TL;DR

This paper investigates how boundary geometry in bent waveguides can trap quantum particles, revealing boundary-induced bound states through conformal mapping and WKB analysis, with implications for quantum confinement in open structures.

Contribution

It introduces a conformal mapping approach to analyze bound states in bent waveguides, simplifying mode coupling considerations and demonstrating boundary effects without classical analogs.

Findings

Bound states arise solely from boundary geometry.

Conformal coordinates simplify the mode coupling analysis.

Boundary effects induce quantum trapping without classical counterparts.

Abstract

Is it possible to trap a quantum particle in an open geometry? In this work we deal with the boundary value problem of the stationary Schroedinger (or Helmholtz) equation within a waveguide with straight segments and a rectangular bending. The problem can be reduced to a one dimensional matrix Schroedinger equation using two descriptions: oblique modes and conformal coordinates. We use a corner-corrected WKB formalism to find the energies of the one dimensional problem. It is shown that the presence of bound states is an effect due to the boundary alone, with no classical counterpart for this geometry. The conformal description proves to be simpler, as the coupling of transversal modes is not essential in this case.