Communications in Mathematical Sciences

Volume 14 (2016)

Number 7

Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents

Pages: 1821 – 1858

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n7.a3

Authors

Agissilaos Athanassoulis (Department of Mathematics, University of Leicester, United Kingdom)

Theodoros Katsaounis (Computer Electrical and Mathematical Sciences Engineering, King Abdullah Univ. of Science and Technology (KAUST), Thuwal, Saudi Arabia; and IACM–FORTH, Heraklion, Greece)

Irene Kyza (Division of Mathematics, University of Dundee, Scotland, United Kingdom; and Institute of Applied and Computational Mathematics – FORTH, Vassilika Vouton, Heraklion-Crete, Greece)

Abstract

Semiclassical asymptotics for Schrödinger 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 $-\lvert x \rvert$, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for $-\lvert x \rvert$ 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.

Keywords

semiclassical limit for rough potential, Wigner transform, multivalued flow, selection principle, <i>a posteriori</i> error control

2010 Mathematics Subject Classification

35B60, 35Q41, 35Q83, 65M12, 65M15, 81S30

Full Text (PDF format)