  In this paper, we improve slightly Erd\'elyi's version of the stationary
phase method by replacing the employed smooth cut-off function by a
characteristic function, leading to more precise remainder estimates. We
exploit this refinement to study the time-asymptotic behaviour of the solution
of the free Schr\"odinger equation on the line, where the Fourier transform of
the initial data is compactly supported and has a singularity. We obtain
uniform estimates of the solution in space-time regions which are
asymptotically larger than any space-time cones. Moreover we expand the
solution with respect to time on the boundaries of the above regions, showing
the optimality of the decay rates of the estimates.
