Advertisement

Spectral Theory on Manifolds

  • David Borthwick
Chapter
  • 217 Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 284)

Abstract

This chapter gives a brief introduction to the spectral theory of graphs. The primary focus is on quantum graphs consisting of the Laplacian operator acting on a metric graph.

References

  1. 8.
    Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)CrossRefGoogle Scholar
  2. 9.
    Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics, vol. 194. Springer, Berlin (1971)CrossRefGoogle Scholar
  3. 14.
    Borthwick, D.: Spectral Theory of Infinite-Area Hyperbolic Surfaces. Progress in Mathematics, vol. 318, 2nd edn. Birkhäuser/Springer, Cham (2016)Google Scholar
  4. 15.
    Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178, 501–508 (1981)MathSciNetCrossRefGoogle Scholar
  5. 16.
    Brooks, R.: Inverse spectral geometry. In: Progress in Inverse Spectral Geometry. Trends in Mathematics, pp. 115–132. Birkhäuser, Basel (1997)CrossRefGoogle Scholar
  6. 17.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston (1992)zbMATHGoogle Scholar
  7. 19.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic, London (1984). Including a chapter by Randol, B, With an appendix by Dodziuk, J.Google Scholar
  8. 21.
    Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)MathSciNetCrossRefGoogle Scholar
  9. 23.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)Google Scholar
  10. 26.
    do Carmo, M.P.A.: Riemannian Geometry. Mathematics: Theory and Applications. Birkhäuser, Boston (1992). Translated from the second Portuguese edition by Francis FlahertyCrossRefGoogle Scholar
  11. 27.
    Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46, 497–503 (1979)MathSciNetCrossRefGoogle Scholar
  12. 30.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)Google Scholar
  13. 33.
    Gaffney, M.P.: The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. U.S.A. 37, 48–50 (1951)MathSciNetCrossRefGoogle Scholar
  14. 39.
    Guillemin, V., Pollack, A.:Differential Topology. Prentice-Hall, Englewood Cliffs (1974)Google Scholar
  15. 43.
    Heinonen, J.: Lectures on Lipschitz analysis. Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100. University of Jyväskylä, Jyväskylä (2005)Google Scholar
  16. 47.
    Iwaniec, H.: Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics, vol. 53, 2nd edn. American Mathematical Society, Providence (2002)Google Scholar
  17. 52.
    Klingenberg, W.P.A.: Riemannian Geometry. De Gruyter Studies in Mathematics, vol. 1, 2nd edn. Walter de Gruyter, Berlin (1995)Google Scholar
  18. 55.
    Lablée, O.: Spectral Theory in Riemannian Geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich (2015)CrossRefGoogle Scholar
  19. 57.
    Lee, J.M.: Riemannian Manifolds. An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997)Google Scholar
  20. 58.
    Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2003)CrossRefGoogle Scholar
  21. 62.
    Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949)MathSciNetCrossRefGoogle Scholar
  22. 66.
    Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics, vol. 171, 3rd edn. Springer, Berlin (2016)CrossRefGoogle Scholar
  23. 75.
    Roelcke, W.: Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen. Math. Nachr. 21, 131–149 (1960)MathSciNetCrossRefGoogle Scholar
  24. 81.
    Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. In: Proceedings of the Conference on Lecture Notes in Geometry and Topology, I. International Press, Cambridge (1994)zbMATHGoogle Scholar
  25. 92.
    Venkov, A.B.: Spectral Theory of Automorphic Functions and Its Applications. Kluwer Academic Publishers, Dordrecht (1990)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • David Borthwick
    • 1
  1. 1.Department of MathematicsEmory UniversityAtlantaUSA

Personalised recommendations