Explicit error estimates for the stationary phase method I: The influence of amplitude singularities
arXiv:1412.5789
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.
This paper is part I of a series of two articles. The title of the second part is: "Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points"