Quantum confinement for the curvature Laplacian $− \Delta + cK$ on 2D-almost-Riemannian manifolds
Résumé
Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane.
They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel.
Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a 1D embedded
submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of
these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the
Schr\"{o}dinger equation (with the Laplace-Beltrami operator $\Delta$) cannot. This is due to the fact that (under a natural
compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the
manifold without the singular set. In the literature such phenomenon is called quantum confinement.
In this paper we study the self-adjointness of the curvature Laplacian, namely
$-\Delta+cK$, for $c\in(0,1/2)$ (here $K$ is the Gaussian curvature), which originates
in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization).
We prove that there is no quantum confinement for this type of operators.
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