Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-30T19:40:19.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Scattering operator and Eisenstein integral for Kleinian groups

Published online by Cambridge University Press:  24 October 2008

N. Mandouvalos
N. Mandouvalos
Affiliation:
Aristotle University of Thessaloniki, Department of Mathematics, Thessaloniki, Greece
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we study certain aspects of the problem of analytic continuation of the scattering operator and Eisenstein integral which we introduced in [7, 8, 10], for Kleinian groups Γ with exponent of convergence δ(Γ) ≥ 1. The corresponding problem for groups with δ(Γ) < 1 was examined and solved in [9].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 203 - 217
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ahlors, L. W.. Finitely generated Kleinian groups. Amer. J. Math. 86 (1964), 413–429;CrossRefGoogle Scholar
Ahlors, L. W.. Finitely generated Kleinian groups. Amer. J. Math. 86 (1965), 759.CrossRefGoogle Scholar
[2]Elstrodt, J.. Die Resoirente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I. Math. Ann. 203 (1973), 295–330CrossRefGoogle Scholar
II. Math. Z. 132 (1973), 99–134CrossRefGoogle Scholar
III. Math. Ann. 208 (1974), 99–132.CrossRefGoogle Scholar
[3]Fay, J. D.. Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math. 293/94 (1977), 43–203.Google Scholar
[4]Kubota, T.. Elementary Theory of Eisensiein Series (Halsted Press, 1973).Google Scholar
[5]Langlands, R.. On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Math. vol. 544 (Springer-Verlag, 1977).Google Scholar
[6]Lax, P. D. and Phillips, R.. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46 (1982), 280–350.CrossRefGoogle Scholar
[7]Mandouvalos, N.. The theory of Eisenstein series and spectral theory for Kleinian groups. Ph.D. thesis, Cambridge University (1983).Google Scholar
[8]Mandouvalos, N.. The theory of Eisenstein series for Kleinian groups. In Proceedings of the AMS Conference on the Selberg Trace Formula and Relaled Topics. Contemporary Math. vol. 53 (Birkhäuser, 1986), pp. 357–370.Google Scholar
[9]Mandouvalos, N.. Spectral theory and Eisenstein series for Kleinian groups. Proc. London Math. Soc. (3) 57 (1988), 209–238.CrossRefGoogle Scholar
[10]Mandouvalos, N.. Scattering operator, Eisenstein series, inner product formula and ‘Maass-Selberg’ relations for Kleinian groups. Memoirs Amer. Math. Soc. v. 78, no. 400, 1989.CrossRefGoogle Scholar
[11]Mazzeo, R. R. and Melrose, R. B.. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75 (1987), 260–310.CrossRefGoogle Scholar
[12]Patterson, S. J.. The Laplacian operator on a Riemann surface. II. Compositio Math. 32 (1976), 71–112.Google Scholar
[13]Patterson, S. J.. Measures on limit sets of Kleinian groups. In Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Note Series no. 111 (Cambridge University Press, 1987), pp. 281–323.Google Scholar
[14]Patterson, S. J.. The Selberg zeta function of a Kleinian group. In Number Theory, Trace Formulas and Discrete Groups (Academic Press, 1989), pp. 409–441.Google Scholar
[15]Perry, P. A.. The Laplace operator on a hyperbolic manifold, I. Spectral theory and scattering theory. J. Funct. Anal. 75 (1987), 161–187; II. (Preprint.)CrossRefGoogle Scholar
[16]Phillips, R., Wiskott, B. and Woo, A.. Scattering theory for the wave equation on a hyperbolic manifold. J. Funct. Anal. 74 (1987), 346–398.CrossRefGoogle Scholar