We study spectral properties of the Laplace-Beltrami operator on a class of relevant almost-Riemannian manifolds, namely the Grushin structures onthe cylinder and on the sphere. As for general almost-Riemannian structures , 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 connectedcomponent of the manifold without the singular set.
We will present explicit descriptions of the spectrum, of the eigenfunctions and their properties. In particular in both cases we obtain a Weyl law with dominant term $E\log E$. We will also discuss the drastic effect that an Aharonov-Bohm magnetic potential has on the spectral properties.
Finally, in the last part of the talk we will consider some other generalized Riemannian structures including conic and anti-conic type manifolds . In this case, the Aharonov-Bohm magnetic potential can affect the self-adjointness of the Laplace-Beltrami operator, altering the nature of the communication between the two sides of the singularity.
This is a joint work with U. Boscain and M. Seri.
 Boscain, U., Laurent, C., The Laplace-Beltrami operator in almost-Riemannian Geometry, Ann. Inst. Fourier, to appear.
 Boscain, U., Prandi, D., The heat and Schrödinger equations on conic and anticonic-type surfaces, preprint.