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Quantum confinement for the curvature Laplacian $− \frac{1}{2} ∆ + cK$ on 2D-almost-Riemannian manifolds

Abstract : 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 structure is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator ∆) 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 $− 1 2 ∆ + cK$, for c > 0 (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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https://hal.inria.fr/hal-02996832
Contributor : Eugenio Pozzoli <>
Submitted on : Monday, November 9, 2020 - 6:17:41 PM
Last modification on : Thursday, November 12, 2020 - 11:42:35 AM

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Ivan Beschastnyi, Ugo Boscain, Eugenio Pozzoli. Quantum confinement for the curvature Laplacian $− \frac{1}{2} ∆ + cK$ on 2D-almost-Riemannian manifolds. 2020. ⟨hal-02996832⟩

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