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Scattering poles for asymptotically hyperbolic manifolds


Authors: David Borthwick and Peter Perry
Journal: Trans. Amer. Math. Soc. 354 (2002), 1215-1231
MSC (2000): Primary 58J50, 35P25; Secondary 47A40
DOI: https://doi.org/10.1090/S0002-9947-01-02906-3
Published electronically: October 26, 2001
MathSciNet review: 1867379
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Abstract: For a class of manifolds $X$ that includes quotients of real hyperbolic $ (n+1)$-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on $X$ coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for $X$. In order to carry out the proof, we use Shmuel Agmon's perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.


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  • 1. S. Agmon, A perturbation theory for resonances. Comm. Pure Appl. Math. 51 (1998), 1255-1309. MR 99i:47023; erratum MR 2000g:47010
  • 2. D. Borthwick, Scattering theory and deformations of asymptotically hyperbolic metrics. Preprint, 1997.
  • 3. R. Froese, P. Hislop, P. Perry, A Mourre estimate and related bounds for hyperbolic manifolds with cusps of non-maximal rank. J. Funct. Anal. 98 (1991), 292-310. MR 92h:58198
  • 4. I. C. Gohberg, E. I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché. Math. U. S. S. R. Sbornik 84 (1971), 607-629. MR 47:2409
  • 5. L. Guillopé, M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces. J. Funct. Anal. 129 (1995), 364-389. MR 96b:58116
  • 6. L. Guillopé, M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant curvature. Asymptotic Anal. 11 (1995), 1-22. MR 96h:58172
  • 7. L. Guillopé, M. Zworski, Scattering asymptotics for Riemann surfaces. Ann. Math. 145 (1997), 597-660. MR 98g:58181
  • 8. G. Hagedorn, Link between scattering resonances and dilation-analytic resonances in few-body quantum mechanics. Commun. Math. Phys. 65 (1979), 181-201. MR 80f:81093
  • 9. A. Jensen, Local distortion technique, resonances, and poles of the $S$-matrix. J. Math. Anal. Appl. 59 (1977), 505-513. MR 55:14017
  • 10. A. Jensen, Resonances in an abstract analytic scattering theory. Ann. Inst. Henri Poincaré 33 (1980), 209-223. MR 82b:47007
  • 11. M. Joshi, A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds. Acta. Math. 184 (2000), 41-86. CMP 2000:12
  • 12. T. Kato, Perturbation Theory for Linear Operators. Berlin, Heidelberg, New York: Springer-Verlag, 1976. MR 53:11389
  • 13. F. Klopp, M. Zworski, Generic simplicity of resonances. Helv. Phys. Acta 68 (1995), 531-538. MR 97j:81067
  • 14. P. Lax, R. S. Phillips, Scattering Theory. New York: Academic Press, 1967. MR 36:530
  • 15. P. Lax, R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46, 280-350 (1982). MR 83j:10057
  • 16. P. Lax, R. S. Phillips, Translation representation for automorphic solutions of the non-Euclidean wave equation I, II, III. Comm. Pure. Appl. Math. 37 (1984), 303-328, 37 (1984), 779-813, and 38 (1985), 179-208. MR 86c:58148; MR 86h:58140; MR 86j:58150
  • 17. R. Mazzeo, The Hodge cohomology of a conformally compact metric. J. Diff. Geom. 28 (1988), 309-339. MR 89i:58005
  • 18. R. Mazzeo, Elliptic theory of edge operators. Comm. P. D. E. 16 (1991), 1615-1664. MR 93d:58152
  • 19. R. Mazzeo, Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds. American J. Math. 113 (1991), 25-56. MR 92f:58187
  • 20. R. Mazzeo, R. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75 (1987), 260-310. MR 89c:58133
  • 21. R. B. Melrose, Geometric Scattering Theory. New York, Melbourne: Cambridge University Press, 1995. MR 96k:35129
  • 22. S. J. Patterson, The Selberg zeta-function of a Kleinian group. In Number Theory, Trace Formulas, and Discrete Groups: Symposium in honor of Atle Selberg, Oslo, Norway, July 14-21, 1987, New York, Academic Press, 1989, pp. 409-442. MR 91c:11029
  • 23. S. J. Patterson, P. A. Perry, Divisor of Selberg's zeta function for Kleinian groups. Duke Math. J. 106 (2001), 321-390. CMP 2001:08
  • 24. P. A. Perry, The Laplace operator on a hyperbolic manifold, II. Eisenstein series and the scattering matrix. J. Reine Angew. Math. 398 (1989), 67-91. MR 90g:58138
  • 25. P. A. Perry, The Selberg zeta function and a local trace formula for Kleinian groups. J. Reine Angew. Math. 410 (1990), 116-152. MR 92e:11057
  • 26. M. Reed, B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory. New York, Academic Press, 1979. MR 80m:81085
  • 27. J. Sjöstrand, M. Zworski, Lower bounds on the number of scattering poles. Comm. P. D. E. 18 (1993), 847-857. MR 94h:35198
  • 28. K. Uhlenbeck, Generic properties of eigenfunctions. American J. Math. 98 (1976), 1059-1078. MR 57:4264
  • 29. M. Zworski, Counting scattering poles. In Proceedings of the Taniguchi International Workshop, Spectral and Scattering Theory, M. Ikawa, ed., Marcel Dekker, New York, Basel, Hong Kong, 1994. MR 95i:35210
  • 30. M. Zworski, Dimension of the limit set and density of resonances for convex co-compact hyperbolic quotients. Inventiones Math. 136 (1999), 353-409. CMP 99:12
  • 31. M. Zworski, Resonances in physics and geometry. Notices Amer. Math. Soc. 46 (1999), no. 3, 319-328. MR 2000d:58051

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Additional Information

David Borthwick
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: davidb@mathcs.emory.edu

Peter Perry
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506–0027
Email: perry@ms.uky.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02906-3
Keywords: Scattering resonances, hyperbolic manifolds
Received by editor(s): March 19, 1999
Received by editor(s) in revised form: June 28, 2001
Published electronically: October 26, 2001
Additional Notes: Supported in part by NSF grant DMS-9796195 and by an NSF Postdoctoral Fellowship.
Supported in part by NSF grant DMS-9707051
Article copyright: © Copyright 2001 American Mathematical Society

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