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Dynamical Zeta Function for Several Strictly Convex Obstacles

  Published:2008-03-01
 Printed: Mar 2008
  • Vesselin Petkov
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Abstract

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 $\zeta(s)$. Let $\Pi(s)$ 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 $\Pi(s)$ 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 $Z_D(s), \Pi(s)$ has a singularity at $s = s_2$.
Keywords: dynamical zeta function, periodic rays dynamical zeta function, periodic rays
MSC Classifications: 11M36, 58J50 show english descriptions Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
11M36 - Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
58J50 - Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
 

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