We construct a family of Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton-Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.
In the framework of toroidal Pseudodifferential operators on the flat torus T n := (R/2π Z) n we begin by proving the closure under composition for the class ofwe exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on T n × R n written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space H 1 (T n ; C) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton-Jacobi equation 12 |P + ∇ x v(P, x)| 2 + V (x) =H (P) for P ∈ Z n with > 0, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of P + ∇ x v.
This paper deals with the phase space analysis for a family of Schr¨odinger\ud
eigenfunctions ψ on the flat torus by the semiclassicalWave Front\ud
Set. We study those ψ such that WF is contained in the graph of the gradient\ud
of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that\ud
the semiclassical Wave Front Set of such Schr¨odinger eigenfunctions is stable under\ud
viscous perturbations of Mean Field Game kind. These results provide a further viewpoint,\ud
and in a wider setting, of the link between the smooth invariant tori of Liouville\ud
integrable Hamiltonian systems and the semiclassical localization of Schr¨odinger\ud
eigenfunctions on the torus
For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators of the dynamics. On the one hand, they provide the same results on applications than other popular indicators; on the other hand, their PDE based definition -that we show robust under suitable perturbations -allows one to study them using the traditional tools of PDE environment. In particular, we formulate this approximating device in the Lyapunov exponents framework and we compare the operative use of them to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.
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