Zeta Functions in Scattering Problems Evaluated on the Real Wave Number Axis

Using the two-dimensional 3-disk system (in the A 1 representation) we compare the cluster phase shifts of the exact quantum mechanical problem with the corresponding cluster phase shift of the following semi-classical zeta functions: (a) the Gutzwiller-Voros zeta function, (b) the dynamical zeta function and (c) the quasi-classical determinant. Furthermore we show results for the squared moduli of the quantum mechanical and semi-classical determinants on the real wave number axis. As the cluster phase shifts are in principle measurable quantities, even experimentalists can now decide which of the three choices is the best.