[La trace du noyau de la chaleur et l’existence de résonances de diffusion pour les potentiels bornés]
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.
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Keywords: Scattering, resonances, heat trace
Mot clés : Diffusion, résonances, noyau de la chaleur
@article{AIF_2016__66_2_455_0,
author = {Smith, Hart F. and Zworski, Maciej},
title = {Heat traces and existence of scattering resonances for bounded potentials},
journal = {Annales de l'Institut Fourier},
pages = {455--475},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {66},
number = {2},
year = {2016},
doi = {10.5802/aif.3016},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3016/}
}
TY - JOUR AU - Smith, Hart F. AU - Zworski, Maciej TI - Heat traces and existence of scattering resonances for bounded potentials JO - Annales de l'Institut Fourier PY - 2016 SP - 455 EP - 475 VL - 66 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3016/ DO - 10.5802/aif.3016 LA - en ID - AIF_2016__66_2_455_0 ER -
%0 Journal Article %A Smith, Hart F. %A Zworski, Maciej %T Heat traces and existence of scattering resonances for bounded potentials %J Annales de l'Institut Fourier %D 2016 %P 455-475 %V 66 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3016/ %R 10.5802/aif.3016 %G en %F AIF_2016__66_2_455_0
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. http://www.numdam.org/articles/10.5802/aif.3016/
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