These notes summarize the papers [8, 9] on the analysis of resolvent, Eisenstein series and scattering operator for geometrically finite hyperbolic quotients with rational non-maximal rank cusps. They complete somehow the talk given at the PDE seminar of Ecole Polytechnique in october 2005.
@article{SEDP_2005-2006____A3_0,
author = {Guillarmou, Colin},
title = {Scattering and resolvent on geometrically finite hyperbolic manifolds with rational cusps},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:3},
pages = {1--15},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2005-2006},
mrnumber = {2276069},
language = {en},
url = {http://www.numdam.org/item/SEDP_2005-2006____A3_0/}
}
TY - JOUR AU - Guillarmou, Colin TI - Scattering and resolvent on geometrically finite hyperbolic manifolds with rational cusps JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:3 PY - 2005-2006 SP - 1 EP - 15 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2005-2006____A3_0/ LA - en ID - SEDP_2005-2006____A3_0 ER -
%0 Journal Article %A Guillarmou, Colin %T Scattering and resolvent on geometrically finite hyperbolic manifolds with rational cusps %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:3 %D 2005-2006 %P 1-15 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2005-2006____A3_0/ %G en %F SEDP_2005-2006____A3_0
Guillarmou, Colin. Scattering and resolvent on geometrically finite hyperbolic manifolds with rational cusps. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 3, 15 p. http://www.numdam.org/item/SEDP_2005-2006____A3_0/
[1] U. Bunke, M. Olbrich, Scattering theory for geometrically finite groups, Arxiv: math.DG/9904137.
[2] R. Froese, P. Hislop, P. Perry, The Laplace operator on hyperbolic three-manifolds with cusps of non-maximal rank, Invent. Math. 106 (1991), 295-333. | MR | Zbl
[3] I.M. Gelfand, G.E. Shilov, Generalized functions, Vol 1, Academic Press, New-york and London, 1964. | MR | Zbl
[4] R. Graham Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo, Ser.II, Suppl. 63 (2000), 31-42. | MR | Zbl
[5] C.R. Graham, R. Jenne, L.J. Manson, G.A.J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557-565. | MR | Zbl
[6] C.R. Graham, M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89-118. | MR | Zbl
[7] C. Guillarmou, Resonances and scattering poles on asymptotically hyperbolic manifolds, Math. Res. Letters, 12 (2005), 103-119. | MR | Zbl
[8] C. Guillarmou, Resonances on some of geometrically finite hyperbolic manifolds, to appear Comm. P.D.E. | MR | Zbl
[9] C. Guillarmou, Scattering theory on geometrically finite quotients with rational cusps, submitted.
[10] C. Guillarmou, Generalized Krein formula, determinants and Selberg zeta function in even dimension, preprint Arxiv.
[11] L. Guillopé, M. Zworski Upper bounds on the number of resonances for non-compact complete Riemann surfaces, J. Funct. Anal. 129 (1995), 364-389. | MR | Zbl
[12] L. Guillopé, M. Zworski Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymp. Anal. 11 (1995), 1-22. | MR | Zbl
[13] L. Guillopé, M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. Math. 145 (1997), 597-660. | MR | Zbl
[14] M. Joshi, A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41-86. | MR | Zbl
[15] P. Lax, R. Phillips, The asymptotic distribution of lattice points and noneuclidean spaces, J. Funct. Anal. 46 (1982), no. 3, 280-350. | MR | Zbl
[16] R. Mazzeo , Elliptic theory of differential edge operators. I, Comm. P.D.E. 16 (1991), 1615-1664. | MR | Zbl
[17] R. Mazzeo, Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds, American J. Math. 113 (1991), 25-56. | MR | Zbl
[18] 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 | Zbl
[19] R. Mazzeo, R. Melrose, Pseudo-differential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), no.4, 833-866. | MR | Zbl
[20] R. Mazzeo, R. Phillips, Hodge theory for hyperbolic manifolds, Duke Math. J. 60 (1990), no. 2, 509-559. | MR | Zbl
[21] R.Mazzeo, A. Vasy, Analytic continuation of the resolvent of the Laplacian on symmetric spaces of non-compact types, J. Funct. Anal. 228 (2005), 311-368. | MR | Zbl
[22] R. Melrose, The Atiyah-Patodi-Singer index theorem (AK Peters, Wellesley, 1993). | MR | Zbl
[23] R. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. Spectral and scattering theory (Sanda, 1992), 85–130, Lecture Notes in Pure and Appl. Math., 161, Dekker, New York, 1994. | MR | Zbl
[24] S. Patterson, P. Perry, The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein., Duke Math. J. 106 (2001) 321-391. | Zbl
[25] P. Perry, Meromorphic continuation of the resolvent for Kleinian Groups, Spectral problems in geometry and arithmetic (Iowa City, IA, 1997), Contemp. Math. 237 (1999), 123-147. | MR | Zbl
[26] P. Perry, The Laplace operator on a hyperbolic manifold II, Eisenstein series and the scattering matrix, J. Reine Angew. Math. 398 (1989) 67-91. | MR | Zbl
[27] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J., 59 (1989), 311-323. | MR | Zbl
