TY - JOUR

T1 - Regularized semiclassical limits

T2 - Linear flows with infinite lyapunov exponents

AU - Athanassoulis, Agissilaos

AU - Katsaounis, Theodoros

AU - Kyza, Irene

N1 - Publisher Copyright:
© 2016 International Press.

PY - 2016

Y1 - 2016

N2 - 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 -|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.

AB - 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 -|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.

KW - A posteriori error control

KW - Multivalued flow

KW - Selection principle

KW - Semiclassical limit for rough potential

KW - Wigner transform

UR - http://www.scopus.com/inward/record.url?scp=84990976852&partnerID=8YFLogxK

U2 - 10.4310/CMS.2016.v14.n7.a3

DO - 10.4310/CMS.2016.v14.n7.a3

M3 - Article

AN - SCOPUS:84990976852

SN - 1539-6746

VL - 14

SP - 1821

EP - 1858

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 7

ER -