Analytic Continuation of the Resolvent of the Laplacian and the Dynamical Zeta Function

Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R , i = 1, . . . , κ0, k0 ≥ 3, and let Rχ(z) = χ(−∆D − z2)−1χ, χ ∈ C∞ 0 (R ), be the cut-off resolvent of the Dirichlet Laplacian −∆D in Ω = RN \ ∪0 i=1Ki. We prove that there exists σ1 < s0 such that Z(s) is analytic for Re(s) ≥ σ1 and the cut-off resolvent Rχ(z) has an analytic continuation for Im(z) < −σ1, |Re(z)| ≥ C > 0.