Heat traces and existence of scattering resonances for bounded potentials
[La trace du noyau de la chaleur et l’existence de résonances de diffusion pour les potentiels bornés]
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 455-475.

Nous montrons qu’en dimensions impaires, un potentiel borné, à support compact et à valeurs réelles, présente au moins une résonance de diffusion. En dimension 3 ou plus, ce résultat était connu seulement pour des potentiels suffisamment réguliers. La démonstration est fondée sur un résultat inverse, montrant que la trace régularisée du noyau de la chaleur associé admet un développement asymptotique complet si et seulement si le potentiel est lisse.

We show that, in odd dimensions, any real valued, bounded potential of compact support has at least one scattering resonance. In dimensions 3 and greater this was previously known only for sufficiently smooth potentials. The proof is based on an inverse result, which shows that the regularized trace of the associated heat kernel admits a full asymptotic expansion if and only if the potential is smooth.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3016
Classification : 35P25, 35K08
Keywords: Scattering, resonances, heat trace
Mot clés : Diffusion, résonances, noyau de la chaleur

Smith, Hart F. 1 ; Zworski, Maciej 2

1 Department of Mathematics University of Washington Seattle, WA 98195 (USA)
2 Department of Mathematics University of California Berkeley, CA 94720 (USA)
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Smith, Hart F.; Zworski, Maciej. Heat traces and existence of scattering resonances for bounded potentials. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 455-475. doi : 10.5802/aif.3016. https://aif.centre-mersenne.org/articles/10.5802/aif.3016/

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