Quantum maps from transfer operators

Abstract

The Selberg zeta function ζ 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, ζ s( s) can be exactly ewritten 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.