Potential Analysis

, Volume 41, Issue 2, pp 517–541 | Cite as

Generalized Schrödinger Semigroups on Infinite Graphs

  • Batu Güneysu
  • Ognjen Milatovic
  • Françoise Truc


With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman–Kac-type representations for the corresponding semigroups and derive several applications thereof.


Feynman–Kac formula Infinite graph Markov process Schrödinger semigroup Vector bundle 

Mathematics Subject Classifications (2010)

39A12 47D08 


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  1. 1.
    Bär, C., Pfäffle, F.: Asymptotic heat kernel expansion in the semi-classical limit. Commun. Math. Phys. 294, 731–744 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chung, F.R.K., Sternberg, S.: Laplacian and vibrational spectra for homogeneous graphs. J. Graph Theory. 16, 605–627 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Golénia, S.: Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians. arXiv:1106.0658
  4. 4.
    Grummt, R., Kolb, M.: Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds. J. Math. Anal. Appl. 388, 480–489 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Güneysu, B.: Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds. Proc. Amer. Math. Soc. (2013, in press)Google Scholar
  6. 6.
    Güneysu, B.: The Feynman–Kac formula for Schrödinger operators on vector bundles over complete manifolds. J. Geom. Phys. 60, 1997–2010 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Güneysu, B.: On generalized Schrödinger semigroups. J. Funct. Anal. 262, 4639–4674 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Güneysu, B., Keller, M., Schmidt, M.: A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs. arXiv:1301.1304v1
  9. 9.
    Güneysu, B., Pallara, D.: Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below. arXiv:1211.6863
  10. 10.
    Güneysu, B., Post, O.: Path integrals and the essential self-adjointness of differential operators on noncompact manifolds. Math. Z. 275, 331–348 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Huang, X.: On stochastic completeness of weighted graphs. Thesis, Univ. of Bielefeld (2011)Google Scholar
  12. 12.
    Keller, M., Lenz, D.: Dirichlet forms and stochastic completneness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5, 198–224 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kenyon, R.: Spanning forests and the vector bundle Laplacian. Ann. Probab. 39, 1983–2017 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kuwae, K., Takahashi, M.: Kato class measures of symmetric Markov processes under heat kernel estimates. J. Funct. Anal. 250, 86–113 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Milatovic, O., Truc, F.: Essential self-adjointness of Schrödinger operators on vector bundles over infinite graphs. arXiv:1307.1213
  17. 17.
    Norris, J.R.: Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  18. 18.
    Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7, 447–526 (1982)CrossRefzbMATHGoogle Scholar
  19. 19.
    Simon, B.: Functional Integration and Quantum Physics, 2nd edn. AMS Chelsea Publishing, Providence, RI (2005)zbMATHGoogle Scholar
  20. 20.
    Singer, A., Wu, H.-T.: Vector diffusion maps and the connection Laplacian. Commun. Pure Appl. Math. 65, 1067–1144 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5, 109–138 (1996)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Batu Güneysu
    • 1
  • Ognjen Milatovic
    • 2
  • Françoise Truc
    • 3
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Mathematics and StatisticsUniversity of North FloridaJacksonvilleUSA
  3. 3.Institut FourierGrenoble UniversitySaint Martin d’Hères CedexFrance

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