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Title: | Quantum confinement for the curvature Laplacian $-Δ+cK$ on 2D-almost-Riemannian manifolds |
Author: | Beschastnyi, Ivan Boscain, Ugo Pozzoli, Eugenio |
Keywords: | Grushin plane Quantum confinement Almost-Riemannian manifolds Coordinate-free quantization procedures Self-adjointness of the Laplacian Inverse square potential |
Issue Date: | 6-Aug-2021 |
Publisher: | Springer |
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 Schrödinger 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. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/36276 |
DOI: | 10.1007/s11118-021-09946-9 |
ISSN: | 0926-2601 |
Appears in Collections: | CIDMA - Artigos AGG - Artigos DMat - Artigos |
Files in This Item:
File | Description | Size | Format | |
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2011.03300[1].pdf | 307.6 kB | Adobe PDF | View/Open |
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