Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude

Abstract

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( |Im z 0 |/h +h ∞ ) outside the set of trapped trajectories and are O(h ∞ ) 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.

Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

“…In fact, one can prove more directly Lemma 4.3 and Lemma 4.5 by applying the proof of Theorem 2 of [4] to the function w − w.…”

“…Moreover, Theorem 3.1 implies that Π zα,θ = O(h −M ) for θ = νh| ln h| and some M > 0. Then Lemma 5.4 of [4] (see also Proposition 5.1 of [28] in the case of a well in the island) states that…”

Erratum to “Resonance Free Domains for Non Globally Analytic Potentials” Ann. Henri Poincaré 3(4) (2002), 739–756

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Resolvent estimates and local energy decay for hyperbolic equations

“…has been obtained in [2], provided that we have no resonances in a neighborhood of W. It is natural to conjecture that under the condition of Theorem 1.3, the cut-off resolvent R χ (λ ) is bounded uniformly on R for any dimension n ≥ 3.…”