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.
@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/
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