Large-time dynamics for the one-dimensional Schrödinger equation

Abstract

Famous results by Rademacher, Kolmogorov and Paley and Zygmund state that random series on the torus enjoy better Lp bounds that the deterministic bounds. We present a natural extension of these harmonic analysis results to a partial-differential-equations setting. Specifically, we consider the one-dimensional nonlinear harmonic oscillator i∂tu + Δu − |x|2u = |u|r−1u, and exhibit examples for which the solutions are better behaved for randomly chosen initial data than would be predicted by the deterministic theory. In particular, on a deterministic point of view, the nonlinear harmonic oscillator equation is well posed in L2(ℝ) if and only if r ≤ 5. However, we shall prove that, for all nonlinearities |u|r−1u, r > 1, not only is the equation well posed for a large set of initial data whose Sobolev regularity is below L2, but also the flows enjoy very nice large-time probabilistic behaviour.These results are joint work with Laurent Thomann and Nikolay Tzvetkov.