Scattering Theory for the Wave Equation on a Hyperbolic Manifold

  • Ralph Phillips
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 7)

Abstract

This talk is a survey of work done jointly, mainly with Peter Lax [3,4,5] but also with Bettina Wilkott and Alex Woo [10], over the past ten years. It deals with the spectrum of the perturbed (and unperturbed) Laplace-Beltrami operator acting on automorphic functions on an n-dimensional hyperbolic space IHn. The associated discrete subgroup Γ of motions is assumed to have the finite geometric property, but is otherwise unrestricted. This means that the fundamental domain, when derived by the polygonal method, has a finite number of sides; its volume may be finite or infinite and it can have cusps of arbitrary rank. With the obvious identifications the fundamental domain can be treated as a manifold, M.

Keywords

Manifold Radon 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Elstrodt, J., Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, Math. Ann., 203 (1973) 295–330MathSciNetMATHCrossRefGoogle Scholar
  2. 1a.
    Elstrodt, J., Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, II, Math. Zeitschr., 132 (1973) 99–134MathSciNetMATHCrossRefGoogle Scholar
  3. 1b.
    Elstrodt, J., Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, III, Math. Ann., 208 (1974) 99–132.MathSciNetMATHCrossRefGoogle Scholar
  4. 2.
    Enss, V., Asymptotic completeness for quantum mechanical potential scattering, Comm. Math. Phys., 61 (1978) 285–291.MathSciNetADSMATHCrossRefGoogle Scholar
  5. 3.
    Lax, P. and Phillips, R., Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, Comm. Pure and Appl. Math. I, 37 (1984) 303–328MathSciNetMATHCrossRefGoogle Scholar
  6. 3a.
    Lax, P. and Phillips, R., Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, Comm. Pure and Appl. Math. II, 37 (1984) 779–813MathSciNetMATHCrossRefGoogle Scholar
  7. 3b.
    Lax, P. and Phillips, R., Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, Comm. Pure and Appl. Math. III, 38 (1985) 179–207.MathSciNetMATHCrossRefGoogle Scholar
  8. 4.
    Lax, P. and Phillips, R., Scattering theory for automorphic functions, Annals of Math. Studies, 87, Princeton Univ. Press, 1976.MATHGoogle Scholar
  9. 5.
    Lax, P., and Phillips R., The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, Jr. of Functional Analysis, 46 (1982) 280–350.MathSciNetMATHCrossRefGoogle Scholar
  10. 6.
    Maass, H., Uber eine neue Art von nichtanalytischen automorphen Functionen und die Bestimmung Dirichletscher Reihen durch Functionalgleichungen, Math. Ann., 121 (1949) 141–183.MathSciNetMATHCrossRefGoogle Scholar
  11. 7.
    Patterson, S.J., The Laplace operator on a Riemann surface, I, Compositio Math., 31 (1975) 83–107MathSciNetMATHGoogle Scholar
  12. 7a.
    Patterson, S.J., The Laplace operator on a Riemann surface, II, Compositio Math., 32 (1976) 71–112MathSciNetMATHGoogle Scholar
  13. 7b.
    Patterson, S.J., The Laplace operator on a Riemann surface, III, Compositio Math., 33 (1976) 227–259.MathSciNetMATHGoogle Scholar
  14. 8.
    Perry, Peter, The Laplace operator on a hyperbolic manifold, I. Spectral and scattering theory, Jr. Funct. Anal., to appear.Google Scholar
  15. 9.
    Phillips, R., Scattering theory for the wave equation with a short range perturbation, Indiana Univ. Math. Jr., 1–31 (1982) 609–639CrossRefGoogle Scholar
  16. 9a.
    Phillips, R., Scattering theory for the wave equation with a short range perturbation, Indiana Univ. Math. Jr., 11–33 (1984) 831–846.CrossRefGoogle Scholar
  17. 10.
    Phillips, Ralph., Wiskott, Bettina and Woo, Alex, Scattering theory for the wave equation on a hyperbolic manifold, Jr. of Punct. Anal., to appear.Google Scholar
  18. 11.
    Roelcke, W., Das eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, Math. Ann., 167 (1966) 292–337MathSciNetCrossRefGoogle Scholar
  19. 11.
    Roelcke, W., Das eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, II, Math. Ann., 168 (1967) 261–324.MathSciNetCrossRefGoogle Scholar
  20. 12.
    Wiskott, Bettina, Scattering theory and spectral representation of short range perturbation in hyperbolic space, Dissertation, Stanford Univ., 1982.Google Scholar
  21. 13.
    Woo, A.C., Scattering theory on real hyperbolic spaces and their compact perturbations, Dissertation, Stanford Univ., 1980.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Ralph Phillips
    • 1
  1. 1.Stanford UniversityUSA

Personalised recommendations