Dynamical Zeta Function for Several Strictly Convex Obstacles

Petkov, Vesselin

Canadian mathematical bulletin, Tome 51 (2008) no. 1, pp. 100-113

Résumé

The behavior of the dynamical zeta function ${{Z}_{D}}(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s)\,=\,\frac{1}{\zeta (s)}$ of the Riemann zeta function $\zeta \left( s \right)$ . Let $\prod \left( s \right)$ be the series obtained from ${{Z}_{D}}(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of ${{Z}_{D}}(s)$ and $\prod \left( s \right)$ close to the line $\Re s={{s}_{2}},$ where ${{s}_{2}}$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions ${{Z}_{D}}(s),$ $\prod \left( s \right)$ has a singularity at $s\,=\,{{s}_{2}}$ .

Détail

DOI : 10.4153/CMB-2008-012-8

Mots-clés : 11M36, 58J50, dynamical zeta function, periodic rays

Bibliographie

[1] [1] Bardos, C., Guillot, J. C. and Ralston, J., La relation de Poisson pour l’équation des ondes dans un ouvert non-borné. Comm. Partial Differantial Equations 7(1982), no. 8905–958. Google Scholar

[2] [2] Burq, N., Controle de l’équation des plaques en présence d’obstacles strictement convexes. Suppl. Bull. Soc. Math. France, 121(1993), Mémoire 55. Google Scholar

[3] [3] Cvitanović, P., Vattay, G. and Wirzba, A., Quantum fluids and classical determinants. In: Classical, Semiclassical and Quantum Dynamics in Atoms, Lecture Notes in Physics 487, Springer, Berlin, pp. 29–62. Google Scholar

[4] [4] Dolgopyat, D., On decay of correlations of Anosov flows. Ann. of Math. 147(1998), no. 2, 357–390. Google Scholar

[5] [5] Guillemin, V. and Melrose, R., The Poisson summation formula for manifolds with boundary. Adv. in Math. 32(1979), no. 3, 204–232. Google Scholar

[6] [6] Ikawa, M., Decay of solutions of the wave equation in the exterior of several strictly convex bodies. Ann. Inst. Fourier (Grenoble), 38(1988), no. 2, 113–146. Google Scholar

[7] [7] Ikawa, M., On the existence of poles of the scattering matrix for several convex bodies. Proc. Japan Acad. Ser. A Math. Sci. 64(1988), no. 4, 91–93. Google Scholar

[8] [8] Ikawa, M., On the distribution of poles of the scattering matrix for several convex bodies. In: Functional-Analytic Methods for Partial Differential Equations, Lecture Notes in Math. 1450, Springer, Berlin, 1990, pp. 210–225. Google Scholar

[9] [9] Ikawa, M., Singular perturbation of symbolic flows and poles of the zeta functions. Osaka J. Math. 27(1990), no. 2, 281–300; Addendum, Osaka J. Math. (1992), 161–174. Google Scholar

[10] [10] Ikawa, M., On scattering by several convex bodies. J. Korean Math. Soc. 37(2000), no. 6, 991–1005. Google Scholar

[11] [11] Lax, P. and Phillips, R., Scattering Theory, 2nd Edition, Pure and Applied Mathematics 26. Academic Press, New York, 1989. Google Scholar

[12] [12] Lin, K. and Zworski, M., Quantum resonances in chaotic scattering. Chem. Phys. Lett. 355(2002), no. 1–2, 201–205. Google Scholar

[13] [13] Melrose, R., Polynomial bound on the distribution of scattering poles. J. Funct. Anal. 53(1983), no. 3, 287–303. Google Scholar

[14] [14] Morita, T., The symbolic representation of billiards without boundary condition. Trans. Amer. Math. Soc. 325(1991), no. 2, 819–828. Google Scholar

[15] [15] Morita, T., Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse. J. Math. Soc. Japan 56(2004), no. 3, 803–831. Google Scholar

[16] [16] Naud, F., Analytic continuation of a dynamical zeta function under a Diophantine condition. Nonlinearity, 14(2001), no. 5, 995–1009. Google Scholar

[17] [17] Parry, W. and Pollicot, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérique, no. 187–188, 1990. Google Scholar

[18] [18] Petkov, V., Analytic singularities of the dynamical zeta function. Nonlinearity, 12(1999), no. 6, 1663–1681. Google Scholar

[19] [19] Petkov, V. and Stoyanov, L., Geometry of Reflecting Rays and Inverse Spectral Problems. Chichester, John Wiley, 1992. Google Scholar

[20] [20] Sjöstrand, J., A trace formula and review of some estimates for resonances. In: Microlocal Analysis and Spectral Theory. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 490, Kluwer, Dordrecht, 1997, 377–437. Google Scholar

[21] [21] Sjöstrand, J. and Zworski, M., Lower bounds on the number of scattering poles. Comm. Partial Differential Equations 18(1993), no. 5–6, 847–857. Google Scholar

[22] [22] Stoyanov, L., Exponential instability for a class of dispersing billiards. Ergodic Theory Dynam. Systems 19(1999), no. 1, 201–226. Google Scholar

[23] [23] Stoyanov, L., Spectrum of the Ruelle operator and exponential decay of correlation flow for open billiard flows. Amer. J. Math. 123 (2001), 715–759. Google Scholar

[24] [24] Stoyanov, L., Ruelle zeta functions and spectra of transfer operators for some Axiom A flows. Preprint. http://www.maths.uwa.edu.au/_stoyanov Google Scholar

[25] [25] Stoyanov, L., Scattering resonances for several small convex bodies and the Lax-Phillips conjecture. To appear in Mem. Amer. Math. Soc. http://www.maths.uwa.edu.au/_stoyanov Google Scholar

[26] [26] Stoyanov, L., Spectra of Ruelle transfer operators for contact flows on basic sets. Preprint, 2007. http://www.maths.uwa.edu.au/_stoyanov Google Scholar

[27] [27] Vodev, G., Sharp bounds on the number of scattering poles for perturbations of the Laplacian. Comm. Math. Phys. 146(1992), no. 1, 205–216. Google Scholar

[28] [28] Voros, A., Spectral functions, special functions and the Selberg zeta function. Commun. Math. Phys. 110(1987), no. 3, 437–465. Google Scholar

[29] [29] Wirzba, A., Quantum mechanics and semi-classics of hyperbolic n-disk scattering systems. Phys. Rep. 309(1999), no. 1–2, 1–116. Google Scholar