The motion of a free particle on a surface of constant negative curvature provides a well-known example of a conservative dynamical system with chaotic classical motion. The authors extend Gutzwiller’s quantum mechanical treatment of such a particular unbound system by the introduction of the time delay. The scattering arises from constraint forces, that is, from motion along geodesics with constant velocity. The associated scattering data are specified in terms of known functions. Among these is Riemann’s zeta-function. The real parts of the poles of the scattering matrix are one half of the imaginary parts of the zeros of the Riemann zeta function. The time delay is expressed in terms of these zeros.