The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus

Abstract

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

A weak KAM approach to the periodic stationary Hartree equation

“…To conclude, we recall that the application of KAM theory or weak KAM theory into the semiclassical Analysis of Schrödinger operators can be given for various problems: like the study of WKB quasimodes, Wigner measures or the asymptotics of the spectrum (see [3,4,6,8,21,23,[34][35][36]43] and references therein). For the use of Fourier Integral Operators and solutions of Hamilton-Jacobi equation to represent the unitary operator solving the quantum dynamics we address to [19,22] and references therein.…”

Mean Field Derivation of DNLS from the Bose–Hubbard Model

“…48 as the parameter h = 1/N → 0 is to describe the essential support (see Theorem 8.16 in [41]). In order to do this, one has to study the semiclassical Wave Front set, also called Frequency Set (see for example [31], or [18] for the periodic setting which is the one of the current paper)…”

Hamilton–Jacobi Homogenization and the Isospectral Problem

“…We underline that a complete study of the link between the effective Hamiltonian, viscosity solutions of Hamilton-Jacobi equation and Schrödinger eigenvalue problem should also involve the phase space analysis of eigenfunctions (and energy quasimodes). In this direction, some preliminary results for the n-dim case have been obtained [18][19][20]. For the time evolution of WKB-type wave functions, we address the reader to the works [21][22][23] (and references therein).…”