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Distribution of Resonances for Asymptotically Euclidean Manifolds
Ist Teil von
Journal of differential geometry, 2000-05, Vol.55 (1), p.43-82
Ort / Verlag
Lehigh University
Erscheinungsjahr
2000
Link zum Volltext
Quelle
Electronic Journals Library
Beschreibungen/Notizen
In this paper we discuss meromorphic continuation of the resolvent and bounds on the
number of resonances for scattering manifolds, a class of manifolds generalizing
Euclidian n-space. Subject to the basic assumption of analyticity near infinity, we
show that resolvent of the Laplacian has a meromorphic continuation to a conic
neighborhood of the continuous spectrum. This involves a geometric interpretation of the
complex scaling method in terms of deformations in the Grauert tube of the manifold. We
then show that the number of resonances (poles of the meromorphic continuation of the
resolvent) in a conic neighborhood of \mathbb{R}_+of absolute value less than
r^2 is \mathcal O(r^n). Under the stronger
assumption of global analyticity and hyperbolicity of the geodesic flow, we prove a finer,
Weyl-type upper bound for the counting function for resonances in small neighborhoods of
the real axis. This estimate has an exponent which involves the dimension of the trapped
set of the geodesic flow.