Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Stefanov, Plamen

Journées équations aux dérivées partielles (2001), article no. 13, 16 p.

We study semiclassical resonances in a box $\Omega \left(h\right)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $𝒯$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^{\#}\left(h\right)$ with discrete spectrum the number of resonances in $\Omega \left(h\right)$ is bounded by the number of eigenvalues of $P^{\#}\left(h\right)$ in an interval a bit larger than the projection of $\Omega \left(h\right)$ on the real line. As an application, we prove a Weyl type estimate of the number of resonances in $\Omega \left(h\right)$ in terms of the measure of $𝒯$. We prove a similar estimate in case of classical scattering by a metric and obstacle.

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