Lossless error estimates for the stationary phase method with applications to propagation features for the Schr\"odinger equation
Abstract
We consider a version of the stationary phase method in one dimension of A. Erd\'elyi, 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 Schr\"odinger 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.
Keywords
Cite
@article{arxiv.1512.05940,
title = {Lossless error estimates for the stationary phase method with applications to propagation features for the Schr\"odinger equation},
author = {F. Ali Mehmeti and F. Dewez},
journal= {arXiv preprint arXiv:1512.05940},
year = {2015}
}
Comments
This paper is a unified version of the two articles "Explicit error estimates for the stationary phase method I: The influence of amplitude singularities" (arXiv:1412.5789) and "Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points" (arXiv:1412.5792). Results and presentation have been slightly improved