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.
Keywords
Cite
@article{arxiv.1412.5792,
title = {Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points},
author = {F. Ali Mehmeti and F. Dewez},
journal= {arXiv preprint arXiv:1412.5792},
year = {2014}
}
Comments
This paper is part II of a series of two articles. The title of the first part is: "Explicit error estimates for the stationary phase method I: The influence of amplitude singularities"