Title :

Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation

Author(s) :

Ali Mehmeti, Felix [Auteur]
Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 [LAMAV]
Dewez, Florent [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Mathematical Models and Methods in Applied Sciences

Pages :

626-662

Publisher :

World Scientific Publishing

Publication date :

2016-05-27

ISSN :

0218-2025

English keyword(s) :

asymptotic expansion
stationary phase method
error estimate
Schrodinger equation
L-infinity-time decay
singular frequency
space-time cone

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

We consider a version of the stationary phase method in one dimension of A. Erdelyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed ...
Show more >We consider a version of the stationary phase method in one dimension of A. Erdelyi, 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 Schrodinger 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 packetsShow less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL