Time decay for solutions of Schrödinger equations with rough and time dependent potentials
I. Rodnianski, W. Schlag
TL;DR
The paper proves dispersive and Strichartz estimates for the 3D Schrödinger equation with both time-independent and time-dependent rough potentials. It develops a robust oscillatory-integral framework that expresses the evolution under a small, time-dependent potential as a convergent Duhamel series whose terms are controlled via resolvent bounds and stationary-phase analysis, including challenging degeneracies. It establishes dispersive decay for small Rollnik/global-Kato potentials, high-energy bounds with an $\varepsilon$ loss, and Strichartz estimates for potentials decaying faster than $|x|^{-2}$, then extends these results to time-dependent potentials in a smallness regime using endpoint Strichartz (Keel–Tao) and functional-calculus techniques. The results bridge classical spectral-perturbation theory with dispersive PDE methods, yielding robust well-posedness and decay results for rough and time-dependent perturbations with broad applicability in quantum dynamics.
Abstract
We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schrödinger equations with potential in three dimensions. Our main focus is on the small rough time-dependent potentials. Examples of such potentials are of the form $V(t,x)=T(t) V_0(x)$, where $T$ is quasiperiodic in time and $V_0$ is essentially an $L^{3/2}$ function of the spatial variables. We also prove the dispersive estimates for small time-independent potentials which belong to the interestion of the Rollnik and global Kato classes. Finally, we settle the question posed by Journe, Soffer, Sogge concerning Strichartz estimates for potentials that decay faster than $|x|^{-2}$.