## Time decay for solutions of Schrödinger equations with rough and time-dependent potentials.(English)Zbl 1063.35035

The authors prove dispersive estimates for the Schrödinger equation in three dimensions, $\frac{1}{i}\partial_t \psi-\Delta\psi +V \psi=0,\quad \psi(s)=f$ where the (small) time-dependent potential satisfies the condition, $\sup_t \| V(t,\cdot)\| _{L^{\frac 3 2}(\mathbb R^3)} +\sup_{x \in \mathbb R^3}\int_{\mathbb R^3}\int_{-\infty}^{\infty} \frac{| V(\hat{\tau},x)| }{| x-y| } \, d\tau \,dy < c_0$ with $$c_0$$ small enough, and where $$V(\hat{\tau},x)$$ denotes the Fourier transform in $$t$$ of $$V(t,x)$$. They prove that, $\left\| \psi(t)\right\| _{\infty}\leq C| t-s| ^-{\frac {3}{2}}\left\| f\right\| _{1},$ for all times $$t,s$$. In the particular case that the potential is time independent they obtain the same estimate under the condition, $\int_{\mathbb R^3} \frac{| V(x)| \, | V(y)| }{| x-y| ^2}\, dx\,dy < (4\pi)^2, \text{ and } \sup_{x\in \mathbb R^3} \int_{R^3}\frac{| V(x)| }{| x-y| }\, dy < 4 \pi.$ They also prove a dispersive estimate with an $$\varepsilon$$-loss without assuming that the potential is small. Furthermore, they prove Strichartz estimates in the case that the potential decays as $$| x| ^{-2-\varepsilon}$$ in three or more dimensions. This solves a problem posed by Journé, Soffer and Sogge.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35J10 Schrödinger operator, Schrödinger equation 35Q40 PDEs in connection with quantum mechanics
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