Large-time dynamics for the one-dimensional Schrödinger equation
Published online by Cambridge University Press: 04 April 2011
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.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 2 , April 2011 , pp. 227 - 251
- Copyright
- Copyright © Royal Society of Edinburgh 2011
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