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

We study resonances near the real axis (jIm zj D O(hN), 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 form also 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 T. We construct a suitable reference operator from P(h) by imposing a microlocal barrier outside T to show that the counting function for those resonances admits an upper bound of Weyl’s type connected with the measure of T. We give an example of system for which this bound is optimal and also prove similar bound in case of classical scattering by obstacle.