Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Vesselin Petkov; Latchezar Stoyanov

doi:10.1016/j.crma.2007.10.019

Partial Differential Equations

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function
[Prolongement analytique de la résolvante du Laplacien et de la fonction zeta dynamique]

Présenté par : Gilles Lebeau

Vesselin Petkov ¹ ; Latchezar Stoyanov ²

¹ Université Bordeaux I, Institut de mathématiques, 351, cours de la Libération, 33405 Talence, France
² University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia

Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 567-572.

Résumé
Abstract

Soit $s_{0} < 0$ l'abscisse de convergence absolue de la fonciton zeta dynamique $Z (s)$ pour des obstacles compacts, disjoints et strictement convexes $K_{i} \subset R^{N}, i = 1, \dots, κ_{0}, κ_{0} ⩾ 3$ et soit $R_{χ} (z) = χ {(- Δ_{D} - z^{2})}^{−1} χ, χ \in C_{0}^{\infty} (R^{N})$ , la résolvante tronquée du Laplacien de Dirichlet $- Δ_{D}$ dans $Ω = \bar{R^{N} ∖ ⋃_{i = 1}^{κ_{0}} K_{i}}$ . On prouve qu'il existe $σ_{2} < s_{0}$ tel que $Z (s)$ est analytique pour $R (s) ⩾ σ_{2}$ et la résolvante tronquée $R_{χ} (z)$ admet un prolongement analytique pour $ℑ (z) < - i σ_{2}, | R (z) | ⩾ C$ .

Let $s_{0} < 0$ be the abscissa of absolute convergence of the dynamical zeta function $Z (s)$ for several disjoint strictly convex compact obstacles $K_{i} \subset R^{N}, i = 1, \dots, κ_{0}, κ_{0} ⩾ 3$ and let $R_{χ} (z) = χ {(- Δ_{D} - z^{2})}^{−1} χ, χ \in C_{0}^{\infty} (R^{N})$ , be the cut-off resolvent of the Dirichlet Laplacian $- Δ_{D}$ in $Ω = \bar{R^{N} ∖ ⋃_{i = 1}^{κ_{0}} K_{i}}$ . We prove that there exists $σ_{2} < s_{0}$ such that $Z (s)$ is analytic for $R (s) ⩾ σ_{2}$ and the cut-off resolvent $R_{χ} (z)$ has an analytic continuation for $ℑ (z) < - i σ_{2}, | R (z) | ⩾ C$ .

Reçu le : 2007-09-24
Accepté le : 2007-10-08
Publié le : 2007-11-07

DOI : 10.1016/j.crma.2007.10.019

Affiliations des auteurs :

Vesselin Petkov ¹ ; Latchezar Stoyanov ²

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     author = {Vesselin Petkov and Latchezar Stoyanov},
     title = {Analytic continuation of the resolvent of the {Laplacian} and the dynamical zeta function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {567--572},
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     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.019},
     language = {en},
}

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Vesselin Petkov; Latchezar Stoyanov. Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 567-572. doi : 10.1016/j.crma.2007.10.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.019/

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