Generalized Solutions and Spectrum for Dirichlet Forms on Graphs

  • Sebastian HaeselerEmail author
  • Matthias Keller
Conference paper
Part of the Progress in Probability book series (PRPR, volume 64)


In the framework of regular Dirichlet forms we consider operators on infinite graphs. We study the connection of the existence of solutions with certain properties and the spectrum of the operators. In particular we prove a version of the Allegretto-Piepenbrink theorem which says that positive (super)-solutions to a generalized eigenvalue equation exist exactly for energies not exceeding the infimum of the spectrum. Moreover we show a version of Shnol’s theorem, which says that existence of solutions satisfying a growth condition with respect to a given boundary measure implies that the corresponding energy is in the spectrum.


Ground State Energy Dirichlet Form Minimum Principle Harnack Inequality Boundary Measure 
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  1. 1.
    W. Allegretto. On the equivalence of two types of oscillation for elliptic operators. Pac. J. Math. 55, 319–328, 1974.MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. Boutet de Monvel, D. Lenz and P. Stollmann, Shnol’s theorem for strongly local Dirichlet forms. Israel J. Math. 173, 189–211, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    S.Y. Cheng, S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 1975, 333–354.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    J. Dodziuk, Difference equations, isoperimetric inequalities and transience of certain random walks, Trans. Am. Math. Soc. 284, 787–794, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    J. Dodziuk, Elliptic operators on infinite graphs, Analysis, Geometry and Topology of Elliptic Operators: Papers in Honor of Krzysztof P. Wojciechowski, World Scientific Pub Co, 353–368 2006.Google Scholar
  6. 6.
    J. Dodziuk, L. Karp, Spectral and function theory for combinatorial Laplacians, Geometry of Random Motion, (R. Durrett, M.A. Pinsky ed.) AMS Contemporary Mathematics, Vol 73, 25–40, 1988.Google Scholar
  7. 7.
    J. Dodziuk, V. Mathai, Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel. Contemp. Math., 398, 69–81, Amer. Math. Soc., Providence, RI, 2006.Google Scholar
  8. 8.
    D. Fischer-Colbrie, R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33(2), 199– 211, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R.L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    M. Fukushima, Y. O-shima, M. Takeda, Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-XGoogle Scholar
  11. 11.
    R.L. Frank, B. Simon, T. Weidl, Eigenvalue bounds for perturbations of Schr¨odinger operators and Jacobi matrices with regular ground states. Comm. Math. Phys. 282(1), 199–208, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, preprint (2009), arXiv:0904.2985, to appear in: J. Reine Angew. Math.Google Scholar
  13. 13.
    M. Keller, D. Lenz, Unbounded Laplacians on graphs: Basic spectral properties and the heat quation, Math. Model. Nat. Phenom. 5, no. 4, 198–224, 2010.Google Scholar
  14. 14.
    P. Kuchment, Quantum graphs II. Some spectral properties of quantum and combinatorialgraphs. J. Phys. A, 38(22), 4887–4900, 2005.MathSciNetzbMATHGoogle Scholar
  15. 15.
    T. Kumagai, Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40(3), 793–818, 2004MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    D. Lenz, P. Stollmann, I. Veseli´c, The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms. Doc. Math. 14, 167–189, 2009.Google Scholar
  17. 17.
    D. Lenz, P. Stollmann, I. Veseli´c, Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms. To appear in: OTAMP 2008 proceedings. (arXiv: 0909.1107)Google Scholar
  18. 18.
    W.F. Moss, J. Piepenbrink, Positive solutions of elliptic equations. Pacific J. Math. 75(1), 219–226, 1978.MathSciNetzbMATHGoogle Scholar
  19. 19.
    B. Mohar, W. Woess, A survey on spectra of infinite graphs. Bull. London Math. Soc. 21(3), 209–234, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    J. Piepenbrink, Nonoscillatory elliptic equations. J. Differential Equations, 15, 541– 550, 1974.MathSciNetzbMATHGoogle Scholar
  21. 21.
    W.E. Pruitt, Eigenvalues of non-negative matrices. Ann. Math. Statist. 35, 1797– 1800, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. xv+396 pp. ISBN: 0-12-585004-2Google Scholar
  23. 23.
    B. Simon, Spectrum and continuum eigenfunctions of Schr¨odinger operators. J. Funct. Anal., 42, 347–355, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    B. Simon, Schr¨odinger semigroups. Bull. Amer. Math. Soc., 7(3), 447–526, 1982. [25] I.E. Shnol’, On the behaviour of the eigenfunctions of Schr¨odinger’s equation. Mat. Sb., 42, 273–286, 1957. erratum 46(88), 259, 1957.Google Scholar
  25. 25.
    P. Stollmann, Caught by disorder: A Course on Bound States in Random Media, volume 20 of Progress in Mathematical Physics. Birkh¨auser 2001.Google Scholar
  26. 26.
    D. Sullivan, Related aspects of positivity in Riemannian geometry. J. Differential Geom. 25(3), 327–351, 1987.MathSciNetzbMATHGoogle Scholar
  27. 27.
    D. Vere-Jones, Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22, 361–386, 1967.MathSciNetzbMATHGoogle Scholar
  28. 28.
    D. Vere-Jones, Ergodic properties of nonnegative matrices. II. Pacific J. Math. 26, 601–620, 1968.MathSciNetzbMATHGoogle Scholar
  29. 29.
    W. Woess, Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0- 521-55292-3Google Scholar
  30. 30.
    R.K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419–1441, 2009.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Mathematical InstituteFriedrich Schiller University JenaJenaGermany

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