Point interactions for 3D sub-Laplacians |
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Authors: | Riccardo Adami Ugo Boscain Valentina Franceschi Dario Prandi |
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Affiliation: | 1. Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy;2. CNRS, Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France;3. Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, via Trieste 63, 35131 Padova, Italy;4. Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France;1. Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75252 Paris, France;2. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, PR China;1. EPFL SB, Station 8, CH-1015 Lausanne, Switzerland;2. Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland;3. IMATI-CNR, via Ferrata 5, I-27100 Pavia, Italy;1. Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain;2. Laboratoire Jacques-Louis Lions, Sorbonne Université & CNRS, France;1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brazil;2. Dipartimento di Matematica, Politecnico di Milano, Italy;1. Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy;2. University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany;1. Department of Mathematics, Temple University, Philadelphia, PA 19122, United States of America;2. Instituto Argentino de Matemática A. P. Calderón, CONICET, Buenos Aires, Argentina |
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Abstract: | In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on is essentially self-adjoint in . A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on is never essentially self-adjoint in , if . We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass. |
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Keywords: | Essential self-adjointness Heisenberg group Sub-Laplacian Point interactions Sub-Riemannian geometry Rotation of molecules |
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