Advertisement

Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 331–348 | Cite as

Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

  • Batu GüneysuEmail author
  • Olaf Post
Article

Abstract

We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on \(\mathsf{C }^{\infty }_0\), and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian \(3\)-manifolds.

Keywords

Vector Bundle Dirac Operator Operator Core Noncompact Riemannian Manifold Clifford Multiplication 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The first author (BG) is indebted to Ognjen Milatovic for many discussions on essential self-adjointness in the past three years, in particular, for bringing the reference [13] into our attention (which helped us to remove an unnecessary assumption from the original version of Theorem 1.1). Both authors kindly acknowledge the financial support given by the SFB 647 “Space–Time–Matter” at the Humboldt University Berlin, where this work has been started.

References

  1. 1.
    Braverman, M., Milatovich, O., Shubin, M.: Essential self-adjointness of Schrödinger-type operators on manifolds. Russian Math. Surv. 57(4), 641–692 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chernoff, P.: Essential self-adjointness of powers of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coulhon, T., Sikora, A.: Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem. Proc. Lond. Math. Soc. (3) 96(2), 507–544 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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, Springer Study Edition, pp. x+319. Springer, Berlin (1987).Google Scholar
  5. 5.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)Google Scholar
  6. 6.
    Dodziuk, J.: Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32(5), 703–716 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dollard, J.D., Friedman, C.N.: Product Integration. Addison-Wesley, Menlo Park (1979)zbMATHGoogle Scholar
  8. 8.
    Driver, B.K., Thalmaier, A.: Heat equation derivative formulas for vector bundles. J. Funct. Anal. 183(1), 42–108 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Enciso, A.: Coulomb systems on Riemannian manifolds and stability of matter. Ann. Henri Poincaré 12, 723–741 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gaffney, M.P.: The conservation property of the heat equation on Riemannian manifolds. Commun. Pure Appl. Math. 12, 1–11 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grigor’yan, A.: Heat kernel and analysis on manifolds. AMS/IP Studies in Advanced Mathematics, vol 47. American Mathematical Society/International Press, Providence/Boston (2009)Google Scholar
  12. 12.
    Grigor’yan, A.: Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Differ. Geom. 45, 33–52 (1997)MathSciNetGoogle Scholar
  13. 13.
    Grummt, R., Kolb, M.: Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds. J. Math. Anal. Appl. 388(1), 480–489 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Güneysu, B.: Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds. Proc. Am. Math. Soc. (2012, in press)Google Scholar
  15. 15.
    Güneysu, B.: On generalized Schrödinger semigroups. J. Funct. Anal. 262, 4639–4674 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Güneysu, B.: Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds. Ann. Henri Poincaré 13, 1557–1573 (2012)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hinz, A., Stolz, G.: Polynomial boundedness of eigensolutions and the spectrum of Schrödinger operators. Math. Ann. 294(2), 195–211 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hsu, E.: Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, vol 38. American Mathematical Society, Providence (2002)Google Scholar
  19. 19.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). Reprint of the 1980 editionzbMATHGoogle Scholar
  20. 20.
    Kato, T.: Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kuwae, K., Takahashi, M.: Kato class measures of symmetric Markov processes under heat kernel estimates. J. Funct. Anal. 250(1), 86–113 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lesch, M.: Essential self-adjointness of symmetric linear relations associated to first order systems. Journées Équations aux Dérivées Partielles (La Chapelle sur Erdre, 2000), Exp. No. X, pp. 18, Univ. Nantes, Nantes (2000)Google Scholar
  23. 23.
    Lieb, E., Seiringer, R.: The stability of matter in quantum mechanics. Cambridge University Press, London (2009)CrossRefzbMATHGoogle Scholar
  24. 24.
    Milatovic, O.: Self-adjointness of Schrödinger-type operators with locally integrable potentials on manifolds of bounded geometry. J. Math. Anal. Appl. 295(2), 513–526 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic Press Inc., London (1975)Google Scholar
  26. 26.
    Shubin, M.: Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds. J. Funct. Anal. 186(1), 92–116 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7(3), 447–526 (1982)CrossRefzbMATHGoogle Scholar
  28. 28.
    Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5(2), 109–138 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für Mathematik, Humboldt-Universität zuBerlinGermany
  2. 2.School of MathematicsCardiff UniversityWalesUK

Personalised recommendations