Microlocalization of resonant states and estimates of the residue of the scattering amplitude

Bony, Jean-François; Michel, Laurent

Journées équations aux dérivées partielles (2003), article no. 2, 12 p.

We obtain some microlocal estimates of the resonant states associated to a resonance $z_0$ of an $h$-differential operator. More precisely, we show that the normalized resonant states are $𝒪(\sqrt{|\mathrm{Im}{z}_0|/h}$ $+h^{\infty })$ outside the set of trapped trajectories and are $𝒪\left(h^{\infty }\right)$ in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such bound in some examples.

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