From spectral to scattering form factor

We propose a novel indicator for chaotic quantum scattering processes, the scattering form factor (ScFF). It is based on mapping the locations of peaks in the scattering amplitude to random matrix eigenvalues, and computing the analog of the spectral form factor (SFF). We compute the spectral and scattering form factors of several non-chaotic systems. We determine the ScFF associated with the phase shifts of the leaky torus, closely related to the distribution of the zeros of Riemann zeta function. We compute the ScFF for the decay amplitude of a highly excited string states into two tachyons. We show that it displays the universal features expected from random matrix theory - a decline, a ramp and a plateau - and is in general agreement with the Gaussian unitary ensemble. It also shows some new features, owning to the special structure of the string amplitude, including a “bump” before the ramp associated with gaps in the average eigenvalue density. The “bump” is removed for highly excited string states with an appropriate state dependent unfolding. We also discuss the SFF for the Gaussian β-ensemble, writing an interpolation between the known results of the Gaussian orthogonal, unitary, and symplectic ensembles.