On the existence of the poles of the scattering matrix for several convex bodies

Divisor of the Selberg zeta function for kleinian groups

“…The relationship between scattering poles and closed geodesies has attracted great interest in the case of obstacle scattering, where Gerard and Ikawa [7,10,11] has shown how closed geodesies (trapped rays) "generate" poles of the scattering operator. An important element in Ikawa's analysis is the dynamical zeta function associated to the geodesies.…”

Distribution of resonances for convex co-compact hyperbolic surfaces

“…For I > 0 let X = {z i-> e^)\HI 2 , that is, X is a hyperbolic cylider with one closed hyperbolic orbit constituting its trapped set. The limit set consists of two points and This bound implies that the density of resonances in logarithmic neighbourhoods of Re s = ^ is at least r. An easy modification of the argument of Ikawa [8] shows also that there are infinitely many resonances in some strip ^ > Re 5 > ^ -a. A little more is true: for large enough strips the density of resonances cannot be less than r 1 " 6 , 6 > 0.…”

Dynamical Zeta Function for Several Strictly Convex Obstacles

“…Following a result of Ikawa [7], [8], the existence of an analytic singularity of Z D (s) implies the existence of δ > 0 such that there are infinite number of poles {z j } j∈N of the scattering matrix S(z) satisfying 0 < ℑz j ≤ δ, ∀j ∈ N and the last property is known as the modified Lax-Phillips conjecture. Another motivation for the analysis of Z D (s) is the folklore conjecture that the singularities of Z D (s) should determine approximatively the scattering poles.…”

“…Here λ j ∈ C are the poles of the scattering matrix S(z) related to the problem (1.5) and the summation is over all poles counted with their multiplicities. We refer to [7], [8], [18], [21] for a more detailed description of this link and to [1], [28], [13], [5], [19], [21] for the trace formulas leading to (1.6).…”