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 -matrix that is unitary for real values of the energy. This paramatrix is the -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.
@incollection{JEDP_2000____A7_0,
author = {R. G. Froese and Peter D. Hislop},
title = {On the distribution of resonances for some asymptotically hyperbolic manifolds},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {7},
pages = {1--16},
year = {2000},
publisher = {Universit\'e de Nantes},
doi = {10.5802/jedp.571},
mrnumber = {2001j:58054},
zbl = {01808697},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.571/}
}
TY - JOUR AU - R. G. Froese AU - Peter D. Hislop TI - On the distribution of resonances for some asymptotically hyperbolic manifolds JO - Journées équations aux dérivées partielles PY - 2000 SP - 1 EP - 16 PB - Université de Nantes UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.571/ DO - 10.5802/jedp.571 LA - en ID - JEDP_2000____A7_0 ER -
%0 Journal Article %A R. G. Froese %A Peter D. Hislop %T On the distribution of resonances for some asymptotically hyperbolic manifolds %J Journées équations aux dérivées partielles %D 2000 %P 1-16 %I Université de Nantes %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.571/ %R 10.5802/jedp.571 %G en %F JEDP_2000____A7_0
R. G. Froese; Peter D. Hislop. On the distribution of resonances for some asymptotically hyperbolic manifolds. Journées équations aux dérivées partielles (2000), article no. 7, 16 p.. doi: 10.5802/jedp.571
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