Spectral analysis and the Aharonov-Bohm~effect on certain almost-Riemannian manifolds

TL;DR

This paper investigates the spectral properties of the Laplace-Beltrami operator on specific almost-Riemannian manifolds, analyzing the effects of Aharonov-Bohm magnetic potentials and providing explicit spectral descriptions.

Contribution

It provides explicit spectral and eigenfunction descriptions for the Laplace-Beltrami operator on Grushin structures and studies the impact of magnetic potentials on self-adjointness and spectral behavior.

Findings

Derived Weyl's law with leading term E log E for the studied manifolds.

Explicitly described eigenfunctions and spectra for the Laplace-Beltrami operator.

Showed that Aharonov-Bohm potentials can significantly alter spectral properties and self-adjointness.

Abstract

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term $E\log E$. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator