Time decay for solutions of Schrödinger equations with rough and time dependent potentials

Details Time decay for solutions of Schrödinger equations with rough and time dependent potentials Time decay for solutions of Schrödinger equations with rough and time dependent potentials

2001-10-09

arxiv.org/abs/math/0110098v2

Mathematics

Analysis of PDEs

corrected typo in the abstract

Scientific paper

We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schr\"odinger 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 $|x|^{-2}$.

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