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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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
18.12.2014
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Subjects | |
Online Access | Get full text |
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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. |
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DOI: | 10.48550/arxiv.1412.5789 |