Sharp upper bounds on the number of resonances near the real axis for trapping systems

We study resonances near the real axis (|Im z| = O(h^N), N > 1) and the corresponding resonant states for semiclassical long range operators P(h). Without a priori assumptions on the distribution or on the multiplicities of the resonances, we show that the truncated resonant states form a family of quasimode states for P(h), stable under small perturbations. As a consequence, they also form a family of quasimode states for any suitably defined (self-adjoint) reference operator P#(h), therefore, those resonances are perturbed eigenvalues of P#(h). Next we show that the semiclassical wave front set of the resonant states is contained in the set of trapped directions [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. We construct a suitable reference operator from P(h) by imposing a microlocal barrier outside [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] to show that the counting function for those resonances admits an upper bound of Weyl's type connected with the measure of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]. We give an example of a system for which this bound is optimal and also prove similar bounds in case of classical scattering by obstacle.