Geodesic
Point interactions for 3D sub-Laplacians
Adami, Riccardo1 ; Boscain, Ugo2 ; Franceschi, Valentina3 ; Prandi, Dario4
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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In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0∈M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0∞(M∖{q0}) is essentially self-adjoint in L2(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 C0∞(N∖{q0}) is never essentially self-adjoint in L2(N), if dim⁡N≤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.

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DOI : 10.1016/j.anihpc.2020.10.007
Keywords: Essential self-adjointness, Heisenberg group, Sub-Laplacian, Point interactions, Sub-Riemannian geometry, Rotation of molecules

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 
@article{AIHPC_2021__38_4_1095_0,
     author = {Adami, Riccardo and Boscain, Ugo and Franceschi, Valentina and Prandi, Dario},
     title = {Point interactions for {3D} {sub-Laplacians}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1095--1113},
     publisher = {Elsevier},
     volume = {38},
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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

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