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

1 GECO - Geometric Control Design
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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. As for general almost-Riemannian structures (under certain technical hypothesis), the singular set acts as a barrier for the evolution of the heat and of a quantum particle, although geodesics can cross it. This is a consequence of the self-adjointness of the Laplace-Beltrami operator on each connected component of the manifolds without the singular set. We get explicit descriptions of the spectrum, of the eigenfunctions and their properties. In particular in both cases we get a Weyl law with dominant term $E\log E$. We then study the effect of an Aharonov-Bohm non-apophantic magnetic potential that has a drastic effect on the spectral properties. Other generalized 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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01019955
Contributor : Dario Prandi <>
Submitted on : Monday, July 7, 2014 - 3:23:01 PM
Last modification on : Tuesday, December 10, 2019 - 9:24:01 AM

### Citation

Ugo Boscain, Dario Prandi, Marcello Seri. Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds. Communications in Partial Differential Equations, Taylor & Francis, 2016, 41 (1), pp. 32-50. ⟨10.1080/03605302.2015.1095766⟩. ⟨hal-01019955⟩

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