Explicit error estimates for the stationary phase method I: The influence of amplitude singularities

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. We provide a complete proof and we improve the remainder estimates in the case of regular...

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Bibliographic Details
Main Authors Mehmeti, F. Ali, Dewez, F
Format Journal Article
LanguageEnglish
Published 18.12.2014
Subjects
Mathematics - Analysis of PDEs
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Summary: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. 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 Schr\"odinger 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.
DOI:10.48550/arxiv.1412.5789