Advertisement

On geometric quantum confinement in Grushin-type manifolds

  • Matteo Gallone
  • Alessandro MichelangeliEmail author
  • Eugenio Pozzoli
Article
  • 67 Downloads

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Agrachev, A., Barilari, D., Boscain, U.: Introduction to Riemannian and Sub-Riemannian geometry from Hamiltonian viewpoint, SISSA preprint 09/2012/M (2012)Google Scholar
  2. 2.
    Agrachev, A., Boscain, U., Sigalotti, M.: A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete Contin. Dyn. Syst. 20, 801–822 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boscain, U., Laurent, C.: The Laplace–Beltrami operator in almost-Riemannian geometry. Ann. Inst. Fourier (Grenoble) 63, 1739–1770 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boscain, U., Prandi, D.: Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces. J. Differ. Equ. 260, 3234–3269 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Braverman, M., Milatovich, O., Shubin, M.: Essential selfadjointness of Schrödinger-type operators on manifolds. Uspekhi Mat. Nauk 57, 3–58 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Calin, O., Chang, D.-C.: Sub-Riemannian Geometry. Encyclopedia of Mathematics and its Applications, vol. 126. Cambridge University Press, Cambridge (2009). General theory and examplesCrossRefGoogle Scholar
  7. 7.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, study edn. Springer, Berlin (1987)CrossRefGoogle Scholar
  8. 8.
    do Carmo, M. P.: Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis FlahertyGoogle Scholar
  9. 9.
    Franceschi, V., Prandi, D., Rizzi, L.: On the essential self-adjointness of singular sub-Laplacians. arXiv:1708.09626 (2017)
  10. 10.
    Gaffney, M.P.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. Math. (2) 60, 140–145 (1954)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hall, B.C.: Quantum Theory for Mathematicians. Graduate Texts in Mathematics, vol. 267. Springer, New York (2013)CrossRefGoogle Scholar
  12. 12.
    Nenciu, G., Nenciu, I.: On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in \(\mathbb{R}^n\). Ann. Henri Poincaré 10, 377–394 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pozzoli, E.: Models of quantum confinement and perturbative methods for point interactions, Master Thesis (2018)Google Scholar
  14. 14.
    Prandi, D., Rizzi, L., Seri, M.: Quantum confinement of non-complete Riemannian manifolds. arXiv:1609.01724 (2016)
  15. 15.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1. Academic Press, New York (1972)zbMATHGoogle Scholar
  16. 16.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1975)zbMATHGoogle Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1978)zbMATHGoogle Scholar
  18. 18.
    Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)CrossRefGoogle Scholar
  19. 19.
    Ward, A.D.: The essential self-adjointness of Schrödinger operators on domains with non-empty boundary. Manuscr. Math. 150, 357–370 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

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

Personalised recommendations