# Quantum confinement for the curvature Laplacian $− \Delta + cK$ on 2D-almost-Riemannian manifolds

1 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
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 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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Journal articles
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https://hal.inria.fr/hal-02996832
Contributor : Eugenio Pozzoli Connect in order to contact the contributor
Submitted on : Thursday, September 30, 2021 - 5:06:03 PM
Last modification on : Wednesday, June 8, 2022 - 12:50:08 PM

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• HAL Id : hal-02996832, version 2

### Citation

Ivan Beschastnyi, Ugo Boscain, Eugenio Pozzoli. Quantum confinement for the curvature Laplacian $− \Delta + cK$ on 2D-almost-Riemannian manifolds. Potential Analysis, Springer Verlag, 2021. ⟨hal-02996832v2⟩

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