On the Essential Self-Adjointness of Singular Sub-Laplacians

Abstract

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that, in the compact case, this criterion implies discreteness of the sub-Laplacian spectrum even though the total volume of the manifold is infinite. As a consequence of our result, the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp’s measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This settles a conjecture formulated by Boscain and Laurent (Ann. Inst. Fourier (Grenoble) 63(5), 1739–1770, 2013), under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.

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Acknowledgments

This research has been supported by the Grant ANR-15-CE40-0018 of the ANR, by the iCODE institute (research project of the Idex Paris-Saclay). The first author has been partially supported by the GNAMPA Indam project “Problemi nonlocali e degeneri nello spazio euclideo” and by “Fondazione Ing. Aldo Gini”, Università degli Studi di Padova. This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research” and from the support to this program from EDF. This work has been partially supported by the ANR project ANR-15-IDEX-02. A proceeding version of this paper appeared in [15], whose last section contains also some remarks on the difficulties arising in presence of tangency points on the singular region.

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Correspondence to Valentina Franceschi.

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Franceschi, V., Prandi, D. & Rizzi, L. On the Essential Self-Adjointness of Singular Sub-Laplacians. Potential Anal 53, 89–112 (2020). https://doi.org/10.1007/s11118-018-09760-w

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Keywords

  • Sub-Laplacian
  • Hörmander-type operators
  • Singular measure
  • Popp’s measure
  • Quantum confinement

Mathematics Subject Classification (2010)

  • Primary: 47B25, 53C17, 58J60
  • Secondary: 35Q40, 81Q10