Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points

Abstract

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.