On the distribution of resonances for some asymptotically hyperbolic manifolds | Zendy

Richard Froese | Zendy; Peter D. Hislop | Zendy

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On the distribution of resonances for some asymptotically hyperbolic manifolds

Author(s) -

Richard Froese,

Peter D. Hislop

Publication year - 2000

Publication title -

journées équations aux dérivées partielles

Language(s) - English

Resource type - Journals

eISSN - 2118-9366

pISSN - 0752-0360

DOI - 10.5802/jedp.571

Subject(s) - mathematics , upper and lower bounds , metric (unit) , pure mathematics , mathematical analysis , dimension (graph theory) , matrix (chemical analysis) , separable space , distribution (mathematics) , laplace operator , manifold (fluid mechanics) , function (biology) , unitary state , law , mechanical engineering , operations management , materials science , evolutionary biology , political science , engineering , economics , composite material , biology

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 5-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.

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