The Selberg Zeta Function and Scattering Poles for Kleiman Groups
نویسندگان
چکیده
In this note we present a polynomial bound on the distribution of poles of the scattering operator for the Laplacian on certain hyperbolic manifolds M of infinite volume. The motivation is to understand more fully the geometry of the poles of the scattering operator. The proof uses the relationship between poles of the scattering operator and zeros of the Selberg zeta function for geodesic flow on M . Recall that the classical Selberg zeta function Z(s) [30] is a meromorphic function which describes the lengths /(y) of closed geodesies y on a compact surface S :
منابع مشابه
Analogues of the Artin Factorization Formula for the Automorphic Scattering Matrix and Selberg Zeta-function Associated to a Kleinian Group
For Kleinian groups acting on hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups, to Kleinian groups, and we give a proof that is simple and extendable to more general groups.
متن کاملResummation of classical and semiclassical periodic-orbit formulas.
The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical and the semiclassical Selberg zeta function have poles. Applying a Padé approximation on the expansions of the full Euler products, as well as on the individ...
متن کاملThe Selberg Trace Formula and Selberg Zeta-Function for Cofinite Kleinian Groups with Finite Dimensional Unitary Representations
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification po...
متن کاملThe Selberg Trace Formula and Selberg Zeta-function for Cofinite Kleinian Groups with Finite-dimensional Unitary Representations
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification po...
متن کاملA New Determinant for Quantum Chaos
Dynamical zeta functions [1], Fredholm determinants [2] and quantum Selberg zeta functions [3, 4] have recently been established as powerful tools for evaluation of classical and quantum averages in low dimensional chaotic dynamical systems [5] [8]. The convergence of cycle expansions [9] of zeta functions and Fredholm determinants depends on their analytic properties; particularly strong resul...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007