JournalsaihpcVol. 38, No. 4pp. 1095–1113

Point interactions for 3D sub-Laplacians

  • Ugo Boscain

    CNRS, Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France
  • Valentina Franceschi

    Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, via Trieste 63, 35131 Padova, Italy
  • Dario Prandi

    Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France
  • Riccardo Adami

    Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy
Point interactions for 3D sub-Laplacians cover
Download PDF

A subscription is required to access this article.

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 q0Mq_{0} \in M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0(M{q0})C_{0}^{\infty }(M \setminus \{q_{0}\}) is essentially self-adjoint in L2(M)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 C0(N{q0})C_{0}^{\infty }(N \setminus \{q_{0}\}) is never essentially self-adjoint in L2(N)L^{2}(N), if dimN3\mathrm{\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.

Cite this article

Ugo Boscain, Valentina Franceschi, Dario Prandi, Riccardo Adami, Point interactions for 3D sub-Laplacians. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 4, pp. 1095–1113

DOI 10.1016/J.ANIHPC.2020.10.007