Heat traces and existence of scattering resonances for bounded potentials
Annales de l'Institut Fourier, Volume 66 (2016) no. 2, p. 455-475

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.

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.

Received : 2015-02-16
Accepted : 2015-05-07
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3016
Classification:  35P25,  35K08
Keywords: Scattering, resonances, heat trace
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     pages = {455-475},
     doi = {10.5802/aif.3016},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 66 (2016) no. 2, pp. 455-475. doi : 10.5802/aif.3016. http://www.numdam.org/item/AIF_2016__66_2_455_0/

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