The scattering matrix for 0th order pseudodifferential operators
Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 2185-2237.

We use microlocal radial estimates to prove the full limiting absorption principle for P, a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions as of Colin de Verdière and Saint-Raymond. We define the scattering matrix for P and show that the scattering matrix extends to a unitary operator on appropriate L 2 spaces. After conjugation with natural reference operators, the scattering matrix becomes a 0th order Fourier integral operator with a canonical relation associated to the bicharacteristics of P. The operator P provides a microlocal model of internal waves in stratified fluids as illustrated in the paper of Colin de Verdière and Saint-Raymond.

Nous utilisons des estimations radiales microlocales pour prouver le principe d’absorption limite complet pour P, un opérateur pseudodifférentiel auto-adjoint d’ordre 0 satisfaisant les hypothèses dynamiques hyperboliques de Colin de Verdière et Saint-Raymond. Nous définissons la matrice de diffusion pour P et montrons que la matrice de diffusion s’étend à un opérateur unitaire sur des espaces L 2 appropriés. Après conjugaison avec des opérateurs de référence naturels, la matrice de diffusion devient un opérateur intégral de Fourier d’ordre 0 avec une relation canonique associée aux bicharactéristiques de P. L’opérateur P fournit un modèle microlocal des ondes internes dans les fluides stratifiés comme illustré dans l’article de Colin de Verdière et Saint-Raymond.

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Accepted:
Published online:
DOI: 10.5802/aif.3570
Classification: 58J40
Keywords: Scattering matrix, zeroth order operator, internal wave.
Mot clés : Matrice de diffusion, opérateur d’ordre zéro, vague interne.
Wang, Jian 1

1 Department of Mathematics University of California Berkeley, CA 94720 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Wang, Jian. The scattering matrix for 0th order pseudodifferential operators. Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 2185-2237. doi : 10.5802/aif.3570. https://aif.centre-mersenne.org/articles/10.5802/aif.3570/

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