Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results

TL;DR

This paper develops analytical and numerical methods to accurately compute energy spectra and wave functions of quantum rings of arbitrary shapes using conformal mapping and variational bounds.

Contribution

It introduces a systematic approach combining conformal mapping, analytical formulas, variational bounds, and an extended conformal collocation method for quantum rings of arbitrary shape.

Findings

Analytical spectrum formulas for circular and Robnik quantum rings

Precise variational bounds for ground states

Numerical solutions for approximately 2000 states

Abstract

We obtain systematic approximations to the states (energies and wave functions) of quantum rings (annular drums) of arbitrary shape by conformally mapping the annular domain to a simply connected domain. Extending the general results of Ref.\cite{Amore09} we obtain an analytical formula for the spectrum of quantum ring of arbirtrary shape: for the cases of a circular annulus and of a Robnik ring considered here this formula is remarkably simple and precise. We also obtain precise variational bounds for the ground state of different quantum rings. Finally we extend the Conformal Collocation Method of \cite{Amore08,Amore09} to the class of problems considered here and calculate precise numerical solutions for a large number of states ($\approx 2000$).