High-frequency propagation for the Schrödinger equation on the torus

Abstract

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schroedinger equation on the standard d-dimensional torus T^{d}. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T^{*}T^{d}. This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit \mu of the Wigner distributions corresponding to solutions to the Schroedinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, \mu follows a propagation law described by a family of density-matrix Schroedinger equations on the periodic geodesics of T^{d}. Finally, we present some connections with the study of the dispersive behavior of the Schroedinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schroedinger equation on T^2 are absolutely continuous with respect to the Lebesgue measure.