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

Abstract

We study spectral properties of the Laplace-Beltrami operator on two relevant almostRiemannian 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 selfadjointness of the Laplace-Beltrami operator.

Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces

“…Let us define the Lagrange parenthesis of u, v : J → R associated to (11) as the bilinear antisymmetric form (11) is regular at the endpoint a 1 if for some (and hence any)…”

“…26 Theorem B.1 (Theorem 13.3.1 in [30]). Let A be the Sturm-Liouville operator on L 2 C (J, w(x)dx) defined in (11). Then n + (A) = n − (A) = #{limit-circle endpoints of J}.…”

From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

“…Despite the prototypical nature of (1.1), the study of analogous problems for Grushin-type operators on more general manifolds often proves to be challenging and not as many results are available. In a few recent works [BFI1,BoPSe,Pe], some attention has been given to 2-step Grushin-type operators on the unit sphere S = {z ∈ R 3 : z 2 1 + z 2 2 + z 2 3 = 1} in R 3 . The operator studied in [BoPSe] is self-adjoint with respect to a measure on the sphere that is singular along the equator E = {z ∈ S : z 3 = 0} and can be thought of the Laplace-Beltrami operator for a certain almost-Riemannian structure on S [BoL].…”

Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

“…Note that if we take β = α + 1 in (14) then we obtain an operator similar to (12) and we expect that close to zero the behaviour of functions in L α+1 and ∆ should be the same.…”

“…If for a generic point q of a sub-Riemannian manifold M we have rank D q = dim M , then we call such a structure almost-Riemannian. In this case the set of points q ∈ M where rank D q < dim M is called the singular set and we will denote it by Z. Almost-Riemannian structures were extensively studied in [1,3,[12][13][14]16,17,40]. Unlike sub-Riemannian manifolds they are equipped with an array of canonical Riemannian objects, such as a curvature or volume.…”