On geometric quantum confinement in Grushin-type manifolds

  • Matteo Gallone
  • Alessandro MichelangeliEmail author
  • Eugenio Pozzoli


We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl’s analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace–Beltrami operator.


Geometric quantum confinement Grushin manifold Geodesically (in)complete Riemannian manifold Laplace–Beltrami operator Almost-Riemannian structure Self-adjoint operators in Hilbert space Weyl’s limit-point limit-circle criterion Constant-fibre direct integral 

Mathematics Subject Classification

35J05 35J10 46N50 47B15 47B25 53C17 53C22 53Z05 



We warmly thank U. Boscain for bringing this problem and the related literature to our attention and for several enlightening discussions on the subject. We also thank for the kind hospitality the Istituto Nazionale di Alta Matematica (INdAM), Rome, where part of this work was carried on. This project is also partially funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement no. 765267.


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Authors and Affiliations

  1. 1.International School for Advanced Studies – SISSATriesteItaly
  2. 2.Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, Équipe CageSorbonne UniversitéParisFrance

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