Regularized semiclassical limits: linear flows with infinite Lyapunov exponents

Bibliography:

A. Athanassoulis, Th. Katsaounis, I. Kyza, Regularized semiclassical limits: linear flows with infinite Lyapunov exponents, Communications in Math Sciences, 14(7):1821-1858, 2016

Authors:

A. Athanassoulis, Th. Katsaounis, I. Kyza

Keywords:

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

Year:

2016

Abstract:

Abstract

Semiclassical asymptotics for Schr ̈odinger 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 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 −|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.

Nonlinear Partial Differential Equations in Fluid and Solid Mechanics

http://appliedpde.kaust.edu.sa

http://appliedpde.kaust.edu.sa

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