Semiclassical expansion of parametric correlations functions of the quantum time delay

We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the two-point correlation function, the form factor K(τ, x, y, M), that depends on the number of open channels M, a non-symmetry breaking parameter x and a symmetry breaking parameter y. Several terms in the Taylor expansion about τ = 0, which depend on all parameters, are shown to be identical to those obtained from random matrix theory