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

This paper is devoted to a detailed study of the behavior of the resonances and resonant states near the real axis. We work mainly in the semi-classical setting but most results can be easily translated into the classical one. By resonances near the real axis we mean resonances in a “box” Ω(h) = [a0, b0] + i[−S(h), 0], where 0 < S(h) = O(h), K ≫ 1. Such resonances may exist only for trapping geometries. We accept the convention here that resonances lie in the lower half-plane. For simplicity, we consider compact perturbation P (h) of the Laplacian only. The basic properties are established in the abstract “black box scattering” setting introduced by Sjöstrand and Zworski [SjZ] (see next section). It is well known that if z(h) is a resonance, then there exists a z(h)-outgoing resonant state u(h) satisfying the equation (P (h)− z(h))u(h) = 0. By [B1], [St3], if −Im z = O(h), then