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- Ts), where Ts 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 Ts. Various properties of the transfer operator are investigated both analytically and numerically for several systems.
- Publication:
-
Physica D Nonlinear Phenomena
- Pub Date:
- August 1993
- DOI:
- 10.1016/0167-2789(93)90199-B
- Bibcode:
- 1993PhyD...67...88B