Local Time-Decay of Solutions to Schrödinger Equations with Time-Periodic Potentials

Let H(t)=−Δ+V(t,x) be a time-dependent Schrödinger operator on L2(R3). We assume that V(t,x) is 2π–periodic in time and decays sufficiently rapidly in space. Let U(t,0) be the associated propagator. For u0belonging to the continuous spectral subspace of L2(R3) for the Floquet operator U(2π,0), we study the behavior of U(t,0)u0as t→∞ in the topology of x-weighted spaces, in the form of asymptotic expansions. Generically the leading term is t−3/2B1u0. Here B1is a finite rank operator mapping functions of xto functions of tand x, periodic in t. If n∈Zis a threshold resonance of the corresponding Floquet Hamiltonian −i∂t+H(t), the leading behavior is t−1/2B0u0. The point spectral subspace for U(2π,0) is finite dimensional. If U(2π 0) φj= \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{ - i2\pi \lambda j} $$ \end{document}φj, then U(t,0)φjrepresents a quasi-periodic solution.