Regularized semiclassical limits: linear flows with infinite Lyapunov exponents

TL;DR

This paper develops refined semiclassical estimates and regularized transport equations for linear Schrödinger equations with singular potentials, especially at saddle points with infinite Lyapunov exponents, extending recent well-posedness results.

Contribution

It introduces a family of refined estimates and regularized transport equations for saddle points with infinite Lyapunov exponents, addressing cases with conical singularities and more singular potentials.

Findings

Regularized transport equations are derived for saddle points with infinite Lyapunov exponents.

Numerical analysis identifies phenomena invalidating regularized transport for certain potentials like -|x|.

Rigorous bounds and conjectures are formulated for the conditions under which regularized transport solutions exist.

Abstract

Semiclassical asymptotics for linear 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.