Regularized semiclassical limits: linear flows with infinite Lyapunov exponents
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Semiclassical asymptotics for Schro¨dinger 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.