Quantum maps from transfer operators (Q1312519)

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 \(\text{det (\textbf{1}}-{\mathbf T}_ s)\), where \({\mathbf T}_ s\) 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 \({\mathbf T}_ s\). Various properties of the transfer operator are investigated both analytically and numerically for several systems.