QUANTUM MAPS FROM TRANSFER OPERATORS

The Selberg zeta function zeta(s)(s) yields an exact relationship between the periodic orbits of a fully chaotic Hamiltonian system (the geodesic flow on surfaces of constant negative curvature) and the corresponding quantum system (the spectrum of the Laplace-Beltrami operator on the same manifold). It was found that for certain manifolds, zeta(s)(s) can be exactly rewritten as the Fredholm-Grothendieck determinant det (1 - T(s)), where T(s) is a generalization of the Ruelle-Perron-Frobenius transfer operator. We present an alternative derivation of this result, yielding a method to find not only the spectrum but also the eigenfunctions of the Laplace-Beltrami operator in terms of eigenfunctions of T(s). Various properties of the transfer operator are investigated both analytically and numerically for several systems.