Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: Boscain, U.; Prandi, D.; Seri, M.
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 1(2016)
• Page Start: 32
• Page End: 50
• Publication Date: 2016-01-02
• Subjects: Aharonov-Bohm effect -- almost-Riemannian geometry -- counting function -- embedded eigenvalues -- Laplace-Beltrami operator -- spectral accumulation -- sub-Riemannian geometry -- Weyl's law; 35P20 -- 53Bxx -- 53C17 -- 58J05; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2015.1095766
• Languages: English
• ISSNs: 0360-5302

Abstract

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 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.