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

Bony, Jean-François; Fujiié, Setsuro; Ramond, Thierry; Zerzeri, Maher

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1351-1406.

On étudie le projecteur spectral associé aux résonances engendrées par le sommet du potentiel d’un opérateur de Schrödinger semiclassique. On démontre d’abord une estimation de la résolvante pour les énergies complexes proches de ces résonances. À l’aide de cette estimation et d’une représentation explicite des états résonants, on prouve que le projecteur spectral admet un développement asymptotique en puissances entières de $h$, dont on donne le terme principal. Ce résultat nous permet alors de calculer le résidu de l’amplitude de diffusion en ces résonances. Finalement, on décrit le comportement en temps grand du groupe de Schrödinger en fonction des résonances.

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrödinger group in terms of these resonances.

Classification : 35B34, 35B38, 35C20, 35P25, 81Q20, 81U20
Keywords: Schrödinger operator, quantum resonances, semiclassical analysis, resolvent estimate
Mot clés : opérateur de Schrödinger, résonances quantiques, analyse semiclassique, estimation de la résolvante

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