Effective quantum dynamics on the M\"obius strip

TL;DR

This paper investigates the spectral properties of the Laplace-Beltrami operator on a M"obius strip as its width tends to zero, deriving an effective flat model with a geometric potential and validating it through analytical and numerical methods.

Contribution

It introduces a new effective flat model for the M"obius strip's quantum dynamics that includes a geometric potential, improving spectral approximation accuracy.

Findings

Spectral properties are well approximated by the effective model.

The effective model's spectrum is explicitly computable using Mathieu functions.

Numerical results confirm the analytical predictions.

Abstract

The Laplace-Beltrami operator in the curved M\"obius strip is investigated in the limit when the width of the strip tends to zero. By establishing a norm-resolvent convergence, it is shown that spectral properties of the operator are approximated well by an unconventional flat model whose spectrum can be computed explicitly in terms of Mathieu functions. Contrary to the traditional flat M\"obius strip, our effective model contains a geometric potential. A comparison of the three models is made and analytical results are accompanied by numerical computations.