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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)
Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics, vol. 194. Springer, Berlin (1971)
Borthwick, D.: Spectral Theory of Infinite-Area Hyperbolic Surfaces. Progress in Mathematics, vol. 318, 2nd edn. Birkhäuser/Springer, Cham (2016)
Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178, 501–508 (1981)
Brooks, R.: Inverse spectral geometry. In: Progress in Inverse Spectral Geometry. Trends in Mathematics, pp. 115–132. Birkhäuser, Basel (1997)
Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston (1992)
Chavel, I.: Eigenvalues in Riemannian Geometry. Academic, London (1984). Including a chapter by Randol, B, With an appendix by Dodziuk, J.
Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)
do Carmo, M.P.A.: Riemannian Geometry. Mathematics: Theory and Applications. Birkhäuser, Boston (1992). Translated from the second Portuguese edition by Francis Flaherty
Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46, 497–503 (1979)
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)
Gaffney, M.P.: The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. U.S.A. 37, 48–50 (1951)
Guillemin, V., Pollack, A.:Differential Topology. Prentice-Hall, Englewood Cliffs (1974)
Heinonen, J.: Lectures on Lipschitz analysis. Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100. University of Jyväskylä, Jyväskylä (2005)
Iwaniec, H.: Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics, vol. 53, 2nd edn. American Mathematical Society, Providence (2002)
Klingenberg, W.P.A.: Riemannian Geometry. De Gruyter Studies in Mathematics, vol. 1, 2nd edn. Walter de Gruyter, Berlin (1995)
Lablée, O.: Spectral Theory in Riemannian Geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich (2015)
Lee, J.M.: Riemannian Manifolds. An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997)
Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2003)
Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949)
Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics, vol. 171, 3rd edn. Springer, Berlin (2016)
Roelcke, W.: Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen. Math. Nachr. 21, 131–149 (1960)
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)
Venkov, A.B.: Spectral Theory of Automorphic Functions and Its Applications. Kluwer Academic Publishers, Dordrecht (1990)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Borthwick, D. (2020). Spectral Theory on Manifolds. In: Spectral Theory. Graduate Texts in Mathematics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-38002-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-38002-1_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38001-4
Online ISBN: 978-3-030-38002-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)