Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non‐integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut‐off function employed in the source 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 Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space‐time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space‐time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.