A sharp lower bound for a resonance-counting function in even dimensions
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 579-604

This paper proves sharp lower bounds on a resonance-counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported perturbations of $-\Delta$ on ${ℝ}^{d}$, for example, for the Laplacian for certain metric perturbations on ${ℝ}^{d}$. The proof uses a Poisson formula for resonances, complementary to one proved by Zworski in even dimensions.

L’objet de cette note est de montrer des bornes inférieures optimales pour la fonction de comptage des résonances, dans le cas d’obstacles sur l’espace euclidien en dimension paire ; on ne fait aucune hypothèse de capture du flot de billard extérieur à l’obstacle. Des minorations similaires sont prouvées pour d’autres types de perturbations à support compact sur ${ℝ}^{d}$. La preuve utilise une formule de Poisson pour les résonances, complémentaire d’une formule montrée par Zworski en dimension paire.

Revised : 2016-05-28
Accepted : 2016-06-14
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3092
Classification:  35P25,  58J50,  35P20,  47A40
Keywords: scattering theory, resonance, obstacle, metric
@article{AIF_2017__67_2_579_0,
author = {Christiansen, T. J.},
title = {A sharp lower bound for a resonance-counting function in even dimensions},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {2},
year = {2017},
pages = {579-604},
doi = {10.5802/aif.3092},
language = {en},
url = {http://www.numdam.org/item/AIF_2017__67_2_579_0}
}

Christiansen, T. J. A sharp lower bound for a resonance-counting function in even dimensions. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 579-604. doi : 10.5802/aif.3092. http://www.numdam.org/item/AIF_2017__67_2_579_0/

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