Plamen Stefanov
Sharp upper bounds on the number of resonances near the real axis for trapped systems
(548K, Postscript)
ABSTRACT. We study the resonances for compactly supported perturbations P(h) of the semiclassical Laplacian in the framework of the "black box" scattering. We are interested in resonances in a "box" of height O(h^N) and fixed width, where N>>1. First we prove that all resonant states and "generalized resonant states" decay uniformly outside a neighborhood of the scatterer. Next we prove that they solve (P(h)-z(h))u(h) = O(h^M), N>>M>>1. Instead of working with single resonances, we work with clusters of resonances and we prove that the resonant states are linearly independent in a stable way under small perturbations. This allows us to compare resonances to eigenvalues of certain self-adjoint reference operators with discrete spectrum. In a more specific situation, where P(h) is a h-PDO (for example the semiclassical Schroedinger operator) we show that one can construct the reference operator by modifying the symbol of P(h) outside of the set of the trapped rays, which is separated from the wave front set of the resonant states. This allows us to prove an upper bound on the number of resonances of Weyl's type with "mass" the measure of the trapped rays. In a case of "potential well" surrounded by a non-trapped region, we obtain an asymptotic formula and a resonance-free zone.