Analytic Wave Front Set for Solutions to Schrödinger Equations II -- Long Range Perturbations

Details Analytic Wave Front Set for Solutions to Schrödinger Equations II -- Long Range Perturbations Analytic Wave Front Set for Solutions to Schrödinger Equations II -- Long Range Perturbations

2008-07-31

arxiv.org/abs/0807.4982v1

Mathematics

Analysis of PDEs

34 pages

Scientific paper

This paper is a continuation of a paper by the authors: arXiv:0706.0415, where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of the paper to long-range perturbations (in particular, we can allow potentials growing like $^{2-\varepsilon}$ at infinity). More precisely, we construct a modified quantum free evolution $G_0(-s, hD_z)$ acting on Sj\"ostrand's spaces, and we characterize the analytic wave front set of the solution $e^{-itH}u_0$ of the Schr\"odinger equation, in terms of the semiclassical exponential decay of $G_0(-th^{-1}, hD_z)T u_0$, where $T$ stands for the Bargmann-transform. The result is valid for $t<0$ near the forward non trapping points, and for $t>0$ near the backward non trapping points. It is an extension of a paper by Nakamura (arXiv:math/0605742) to the analytic framework.

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