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

Rodnianski, Igor; Schlag, Wilhelm

Invent. Math. 155, No. 3, 451-513 (2004).

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.

Reviewer: Ricardo Weder (México City)

MSC:

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

Keywords:

Schrödinger equation; dispersive estimates; time-dependent potentials; Strichartz estimates