On the Essential Self-Adjointness of Singular Sub-Laplacians

  • Valentina FranceschiEmail author
  • Dario Prandi
  • Luca Rizzi


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.


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

Mathematics Subject Classification (2010)

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


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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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Authors and Affiliations

  1. 1.Institut Mathématique d’Orsay & Fondation Mathématiques Jacques HadamardUniversité Paris SudOrsayFrance
  2. 2.CNRS, Laboratoire des Signaux & Systémes, CentraleSupélecGif-sur-YvetteFrance
  3. 3.CNRS, Institut FourierUniv. Grenoble AlpesGrenobleFrance

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