On the distribution of resonances for some asymptotically hyperbolic manifolds
Froese, R. G. ; Hislop, Peter D.
Journées équations aux dérivées partielles, (2000), p. 1-16 / Harvested from Numdam

We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute S-matrix that is unitary for real values of the energy. This paramatrix is the S-matrix for a model laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance counting function requires estimates on the growth of the relative scattering phase, and singular values of a family of integral operators.

Publié le : 2000-01-01
@article{JEDP_2000____A7_0,
     author = {Froese, R. G. and Hislop, Peter D.},
     title = {On the distribution of resonances for some asymptotically hyperbolic manifolds},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2000},
     pages = {1-16},
     mrnumber = {2001j:58054},
     zbl = {01808697},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2000____A7_0}
}
Froese, R. G.; Hislop, Peter D. On the distribution of resonances for some asymptotically hyperbolic manifolds. Journées équations aux dérivées partielles,  (2000), pp. 1-16. http://gdmltest.u-ga.fr/item/JEDP_2000____A7_0/

[1] U. Bunke, M. Olbrich, Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group, Ann. Math. 149, 627-689 (1999). | MR 2000f:11110 | Zbl 0969.11019

[2] T. Christiansen, Spectral asymptotics for compactly supported perturbations of the Laplacian on ℝn, Commun. PDE 23, 933-948 (1998). | MR 99j:35157 | Zbl 0912.35115

[3] R. G. Froese, Upper bounds for the resonance counting function for the Schrödinger operators in odd dimensions, Canadian J. Math. 50, 538-546 (1ç1998). | MR 99f:35150 | Zbl 0918.47005

[4] R. G. Froese, P. D. Hislop, Upper bounds for the resonance counting function for some asymptotically hyperbolic manifolds, in preparation.

[5] R. G. Froese, P. D. Hislop, P. A. Perry, A Mourre Estimate and Related Bounds for the Laplace Operator on a Hyperbolic Manifold with Cusps of Nonmaximal Rank, J. Funct. Anal. 98, 292-310 (1991). | MR 92h:58198 | Zbl 0729.58051

[6] R. G. Froese, P. D. Hislop, P. A. Perry, The Laplace operator on hyperbolic Manifolds with Cusps of Non-maximal Rank, Inventiones Math. 106, 295-333 (1991). | MR 93b:11065 | Zbl 0763.58028

[7] C. Gérard, A. Matrinez, Principe d'absorption limite pour les opérateurs de Schrödinger à longue portée, C. R. Acad. Sci. Paris 306, 121-123 (1988). | Zbl 0672.35013

[8] P. D. Hislop, The geometry and spectra of hyperbolic manifolds, Proc. Indian Acad. Sci. (Math. Sci.) 104, 715-776 (1994). | MR 96b:58117 | Zbl 0832.58005

[9] L. Guillopé, M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature at infinity, Asymp. Anal. 11, 1-22 (1995). | MR 96h:58172 | Zbl 0859.58028

[10] L. Guillopé, M. Zworski, Upper bounds on the number of resonances of non-compact Riemann surfaces, J. Func. Anal. 129, 364-389 (1995). | MR 96b:58116 | Zbl 0841.58063

[11] L. Guillopé, M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. of Math. 145, 597-660 (1997). | MR 98g:58181 | Zbl 0898.58054

[12] A. Jensen, High energy resolvent estimates for generalized many-body Schrödinger operators, Publ. RIMS, Kyoto Univ. 25, 155-167 (1989). | MR 90i:35212 | Zbl 0717.35066

[13] M. S. Joshi, A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 2000. | MR 2002g:58052 | Zbl 01541222

[14] P. Lax, R. S. Phillips, Scattering theory for automorphic functions, Ann. Math. Studies 87, Princeton : Princeton University Press, 1976. | MR 58 #27768 | Zbl 0362.10022

[15] N. Mandouvalos, Scattering operator and Eisenstein integral for Kleinian groups, Math. Proc. Cambridge Philos. Soc. 108, 203-217 (1990). | MR 92e:58220 | Zbl 0719.11031

[16] R. Mazzeo, R. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75, 260-310 (1987). | MR 89c:58133 | Zbl 0636.58034

[17] S. J. Patterson, The Laplacian operator on a Riemann surface, I, II, and III, Compositio math. 31, 83-107 (1975) ; 32, 71-112 (1976) ; 33, 227-259 (1976). | Numdam | MR 52 #5575 | Zbl 0321.30020

[18] S. J. Patterson, P. A. Perry, Divisor of the Selberg Zeta function, I. Even Dimensions, to appear in Duke Math. J. 2000.

[19] P. A. Perry, The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix, J. reine. angew. Math. 398, 67-91 (1989). | MR 90g:58138 | Zbl 0677.58044

[20] P. A. Perry, The Selberg zeta function and a local trace formula for Kleinian groups, J. reine. angew. Math. 410, 116-152 (1990). | MR 92e:11057 | Zbl 0697.10027

[21] P. A. Perry, The Selberg Zeta function and scattering poles for Kleinian groups, Bull. Amer. Math. Soc. (N. S.) 24, 327-333 (1991). | MR 92d:58213 | Zbl 0723.11028

[22] P. A. Perry, Poisson formula and lower bounds on resonances for hyperbolic manifolds, preprint 2000.

[23] D. Robert, On the Weyl formula for obstacles, in «Partial differential equations and mathematical physics», 264-285, Progress in Nonlinear Differential Equations and their Applications, Boston : Birhäuser, 1996. | MR 97i:35132 | Zbl 0849.35096

[24] D. Robert, Asymptotique de la phase de diffusion à haute énergie pours les perturbations de second ordre du Laplacien, Ann. Scien. Ecole Norm. Sup. 25, 107-124 (1992). | Numdam | MR 93i:35096 | Zbl 0801.35100

[25] G. Vodev, Sharp bounds on the number of scattering poles for perturbations of the Laplacian, Commun. Math. Phys. 146, 205-216 (1992). | MR 93f:35173 | Zbl 0766.35032

[26] M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, to appear in Inventionnes math. 1999. | MR 2002d:58038 | Zbl 1016.58014

[27] M. Zworski, Counting Scattering Poles, in Spectral and Scattering Theory, M. Ikawa, ed., Marcel Decker, 1994. | MR 95i:35210 | Zbl 0823.35139