  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 $\beta$- ensemble, writing an interpolation between the known
results of the Gaussian orthogonal, unitary, and symplectic ensembles.
