Point interactions for 3D sub-Laplacians

Adami, Riccardo; Boscain, Ugo; Franceschi, Valentina; Prandi, Dario

doi:10.1016/j.anihpc.2020.10.007

Adami, Riccardo ¹ ; Boscain, Ugo ² ; Franceschi, Valentina ³ ; Prandi, Dario ⁴

¹ a Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy
² b CNRS, Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France
³ c Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, via Trieste 63, 35131 Padova, Italy
⁴ d Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France

Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1095-1113.

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Résumé

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point $q_{0} \in M$ exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on $C_{0}^{\infty} (M ∖ {q_{0}})$ is essentially self-adjoint in $L^{2} (M)$ . 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 $C_{0}^{\infty} (N ∖ {q_{0}})$ is never essentially self-adjoint in $L^{2} (N)$ , if $\dim N \leq 3$ . 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.

Reçu le : 2019-12-05
Révisé le : 2020-10-01
Accepté le : 2020-10-22

MR Zbl

DOI : 10.1016/j.anihpc.2020.10.007

Mots-clés : Essential self-adjointness, Heisenberg group, Sub-Laplacian, Point interactions, Sub-Riemannian geometry, Rotation of molecules

Affiliations des auteurs :

Adami, Riccardo ¹ ; Boscain, Ugo ² ; Franceschi, Valentina ³ ; Prandi, Dario ⁴

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     title = {Point interactions for {3D} {sub-Laplacians}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1095--1113},
     publisher = {Elsevier},
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Adami, Riccardo; Boscain, Ugo; Franceschi, Valentina; Prandi, Dario. Point interactions for 3D sub-Laplacians. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1095-1113. doi : 10.1016/j.anihpc.2020.10.007. http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.007/

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