2015/12/18 by F. Ali Mehmeti, Mehmeti, F. Ali, F. Dewez +1
Mathematics · #35B30 #35B40 #35Q41 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 41A80 #Secondary 41A60 #math.AP #msc:35B30 #msc:35B40 #msc:35Q41 #msc:41A60 #msc:41A80
paper · pdf · doi:10.48550/arxiv.1512.05940
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
arxiv created 2015/12/18 · arxiv updated 2015/12/21
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.