[Paper Review] Explicit error estimates for the stationary phase method I: The influence of amplitude singularities

Félix Ali Mehmeti, Florent Dewez|arXiv (Cornell University)|Dec 18, 2014

Advanced Mathematical Physics Problems7 references3 citations

TL;DR

This paper provides a complete proof and improved error estimates for the stationary phase method in one dimension, particularly for oscillatory integrals with non-integer order stationary points and amplitude singularities. It applies the refined method to the free Schrödinger equation, deriving asymptotic expansions with uniformly bounded remainders in space-time cones, revealing how frequency singularities reduce the $L^\infty$-time decay rate below the standard quantum mechanical $t^{-1/4}$ rate in certain regions.

ABSTRACT

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. We provide a complete proof and we improve the remainder estimates in the case of regular amplitude. Then we are interested in 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. Applying the above mentioned method, we obtain asymptotic expansions with respect to time in certain space-time cones, where the coefficients of the remainders are uniformly bounded. These results show the influence of the singularity on the decay.

Motivation & Objective

• To provide a complete and rigorous proof of Erdélyi's stationary phase theorem for non-integer order stationary points and integrable amplitude singularities.
• To improve remainder estimates in the regular amplitude case, achieving better control over error terms.
• To analyze the time-asymptotic behavior of the free Schrödinger equation with initial data whose Fourier transform has a singularity.
• To derive asymptotic expansions for the solution with explicit, uniformly bounded remainder estimates in space-time cones.
• To quantify how the singularity in the frequency domain affects the $L^\infty$-norm decay rate of the solution over time.

Proposed method

• Use of cut-off functions to localize the integral near stationary points and endpoints.
• Application of explicit complex variable substitutions to simplify the phase function in oscillatory integrals.
• Employment of integration by parts to generate asymptotic expansions and derive remainder terms.
• Use of Cauchy's integral theorem to deform integration contours into regions of controlled oscillation, enabling precise error estimation.
• Introduction of a new parameter in the remainder integral to balance singularity and decay, improving estimates in the regular amplitude case.
• Application of the method to the Schrödinger equation via the Fourier solution formula, with analysis in space-time cones defined by critical directions from frequency singularities.

Experimental results

Research questions

• RQ1 How can explicit error estimates be derived for the stationary phase method when the amplitude has integrable singularities and the phase has non-integer order stationary points?
• RQ2 What is the optimal remainder estimate in the case of a regular (non-singular) amplitude, and how can it be improved beyond Erdélyi's original bound?
• RQ3 How does a singularity in the Fourier transform of the initial data affect the long-time $L^\infty$-norm decay rate of the solution to the free Schrödinger equation?
• RQ4 In which regions of space-time does the solution exhibit optimal decay, and can the remainder in the asymptotic expansion be uniformly bounded in these regions?
• RQ5 Can the method be adapted to yield uniform remainder estimates in cones, and what is the resulting decay rate compared to the standard $t^{-1/4}$ quantum mechanical rate?

Key findings

• The paper provides a complete proof of Erdélyi's stationary phase theorem for non-integer order stationary points and amplitude singularities, filling a gap in the literature.
• Improved remainder estimates are obtained for the regular amplitude case by introducing a new parameter in the remainder integral, achieving better decay control.
• For the free Schrödinger equation with initial data having a singularity in its Fourier transform (e.g., $\mathcal{F}u_0(p) = p^{\mu-1}(1-p)\chi_{[0,1]}(p)$ with $\mu \in (0,1)$), the solution exhibits a time decay rate slower than $t^{-1/4}$ in certain space-time cones.
• In regions away from the critical directions (e.g., $x \sim \pm t$), the remainder in the asymptotic expansion is uniformly bounded, allowing for precise leading-order term analysis.
• The leading term of the asymptotic expansion captures the optimal decay rate in these cones, which is diminished by the singularity compared to the $L^2$-based $t^{-1/4}$ rate.
• The method successfully handles the case $\mu \in (0,1/2)$, where the initial data is not in $L^1$ and the decay rate is not covered by classical Strichartz estimates.