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

Petkov, Vesselin; Stoyanov, Latchezar

doi:10.1016/j.crma.2007.10.019

Geodesic

Parcourir par

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

Petkov, Vesselin ¹ ; Stoyanov, Latchezar 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, Tome 345 (2007) no. 10, pp. 567-572.

Voir la notice de l'article provenant de la source Numdam

Abstract (VO)
Résumé

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 $ℑ (z) < - i σ_{2}, | R (z) | ⩾ C$ .

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 $ℑ (z) < - i σ_{2}, | R (z) | ⩾ C$ .

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 :

Petkov, Vesselin ¹ ; Stoyanov, Latchezar ²

@article{CRMATH_2007__345_10_567_0,
     author = {Petkov, Vesselin and Stoyanov, Latchezar},
     title = {Analytic continuation of the resolvent of the {Laplacian} and the dynamical zeta function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {567--572},
     publisher = {Elsevier},
     volume = {345},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.10.019/}
}

TY  - JOUR
AU  - Petkov, Vesselin
AU  - Stoyanov, Latchezar
TI  - Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 567
EP  - 572
VL  - 345
IS  - 10
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.10.019/
DO  - 10.1016/j.crma.2007.10.019
LA  - en
ID  - CRMATH_2007__345_10_567_0
ER  -

%0 Journal Article
%A Petkov, Vesselin
%A Stoyanov, Latchezar
%T Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function
%J Comptes Rendus. Mathématique
%D 2007
%P 567-572
%V 345
%N 10
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.10.019/
%R 10.1016/j.crma.2007.10.019
%G en
%F CRMATH_2007__345_10_567_0

Petkov, Vesselin; Stoyanov, Latchezar. Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. Comptes Rendus. Mathématique, Tome 345 (2007) no. 10, pp. 567-572. doi : 10.1016/j.crma.2007.10.019. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.10.019/

Bibliographie
Cité par

[1] Burq, N. Controle de l'équation des plaques en présence d'obstacles strictement convexes, Mém. Soc. Math. France, Volume 55 (1993), p. 126

[2] Dolgopyat, D. On decay of correlations in Anosov flows, Ann. Math., Volume 147 (1998), pp. 357-390

[3] Ikawa, M. Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, Volume 2 (1988), pp. 113-146

[4] Ikawa, M. Singular perturbations of symbolic flows and the poles of the zeta function, Osaka J. Math., Volume 27 (1990), pp. 281-300

[5] Ikawa, M. On zeta function and scattering poles for several convex bodies, Conf. EDP, Saint-Jean de Monts, SMF, 1994

[6] Ikawa, M. On scattering by several convex bodies, J. Korean Math. Soc., Volume 37 (2000), pp. 991-1005

[7] Parry, W.; Pollicott, M. Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, Volume 187–188 (1990)

[8] Petkov, V.; Stoyanov, L. Geometry of Reflecting Rays and Inverse Spectral Problems, John Wiley & Sons, 1992

[9] Stoyanov, L. Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Amer. J. Math., Volume 123 (2001), pp. 715-759

[10] L. Stoyanov, Spectra of Ruelle transfer operators for contact flows on basic sets, Preprint, 2007

Cité par Sources :