Lossless error estimates for the stationary phase method with applications to propagation features for the Schr\"odinger equation

TL;DR

This paper refines the stationary phase method to handle phase stationary points of non-integer order and amplitude singularities, providing precise error estimates and applying these to analyze the long-time behavior of Schrödinger equation solutions with singular initial data.

Contribution

It introduces an improved stationary phase method accommodating non-integer stationary points and amplitude singularities, with applications to Schrödinger equation propagation analysis.

Findings

Derived asymptotic expansions for Schrödinger solutions in space-time cones.

Established uniform and optimal decay estimates in curved regions.

Showed the impact of frequency bands and singularities on wave propagation.

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.