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

TL;DR

This paper refines the stationary phase method for better error estimates and applies it to analyze the long-time behavior of Schr"odinger equation solutions with singular initial data, achieving uniform decay estimates.

Contribution

It introduces a sharper stationary phase estimate using characteristic functions and applies it to derive optimal decay rates for Schr"odinger solutions with singular initial Fourier transforms.

Findings

Improved error bounds for stationary phase approximation.

Uniform decay estimates for Schr"odinger solutions beyond typical cones.

Optimal decay rate expansions at boundary regions.

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.