We study resonances near the real axis (|Im z| = 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 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.


Additional Information

Print ISSN
pp. 183-224
Launched on MUSE
Open Access
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.