Please use this identifier to cite or link to this item: http://hdl.handle.net/2381/36266
Title: Regularized semiclassical limits: linear flows with infinite Lyapunov exponents
Authors: Athanassoulis, Agisilaos
Kyza, I.
Katsaounis, T.
First Published: 2016
Citation: Communications in Mathematical Sciences (Accepted, In Press)
Abstract: Semiclassical asymptotics for Schrodinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P. L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posterior error controal. Thus rigorous uppen bounds for the asymptotic error on concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM.
DOI Link: TBA
ISSN: 1945-0796
Links: http://intlpress.com/site/pub/pages/journals/items/cms/_home/submissions/index.html
http://hdl.handle.net/2381/36266
Embargo on file until: 1-Jan-10000
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2015, International Press. All rights reserved.
Description: The file associated with this record is under a permanent embargo while publication is In Press in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.
Appears in Collections:Published Articles, Dept. of Mathematics

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