Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Scattering theory for
twisted automorphic functions

Author: Ralph Phillips
Journal: Trans. Amer. Math. Soc. 350 (1998), 2753-2778
MSC (1991): Primary 58G25, 11F72, 35L05
MathSciNet review: 1466954
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with an irreducible unitary representation $ \rho$ and satisfying $u(\gamma z)=\rho(\gamma)u(z)$. The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, $R_-$ and $R_+$, for the solution operator. The scattering operator, which maps $R_-f$ into $R_+f$, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of $\rho$ is one, the elements of the scattering operator cannot vanish. However when $\dim(\rho)>1$ this is no longer the case.

References [Enhancements On Off] (What's this?)

  • [C] Y. Colin de Verdiere, Pseudo-Laplacian. I and II, Ann. Inst. Fourier 32 (1982), 275-286; (1983), 87-113. MR 84k:58221; MR 84k:58222
  • [K] T. Kubota, Elementary theory of Eisenstein series, Wiley, 1973, 110 pp. MR 55:2759
  • [LP1] P. Lax and R. Phillips, Scattering theory, Academic Press, New York, 1967. MR 38:6237
  • [LP2] P. Lax and R. Phillips, Scattering theory for automorphic functions, Ann. of Math. Studies, Vol. 87, Princeton Univ. Press, Princeton, N.J., 1976. MR 58:27768
  • [LP3] P. Lax and R. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), 280-350. MR 83m:10089
  • [LP4] P. Lax and R. Phillips, Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume, Trans. Amer. Math. Soc. 289 (1985), 715-735. MR 86f:11045
  • [PS] R. Phillips and P. Sarnak, Spectrum of Fermat curves, Geom. and Funct. Anal. 1 (1991), 79-146. MR 92a:11061
  • [Se] A. Selberg, Gottingen lectures, 1954. See his collected works, Vol. 1, Springer-Verlag, 1988, pp. 626-674. MR 92k:01083
  • [V] A. Venkov, Spectral theory of automorphic functions, Trudy Math. Inst. Steklov 153 (1981), 172 pp.; English transl. in Proc. Steklov Inst. Math. (1982), no. 4, 163 pp. MR 85j:11060b

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58G25, 11F72, 35L05

Retrieve articles in all journals with MSC (1991): 58G25, 11F72, 35L05

Additional Information

Ralph Phillips
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): January 23, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society