Inventiones mathematicae

, Volume 155, Issue 3, pp 451-513

First online:

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

  • Igor RodnianskiAffiliated withDepartment of Mathematics, Princeton University Email author 
  • , Wilhelm SchlagAffiliated withDivision of Astronomy, Mathematics, and Physics, Caltech Email author 

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In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions
$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$
where V(t,x) is a time-dependent potential that satisfies the conditions
$$\sup_{t}\|V(t,\cdot)\|_{L^{\frac{3}{2}}(\mathbb{R}^3)} + \sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \int_{-\infty}^\infty\frac{|V(\hat{\tau},x)|}{|x-y|}\,d\tau\,dy < c_0.$$
Here c 0 is some small constant and \(V(\hat{\tau},x\) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·)∈L t (L 2 x (ℝ3))∩L 2 t (L 6 x (ℝ3)) for any fL 2(ℝ3) satisfying the dispersive inequality
$$\|\psi(t)\|_{\infty} \le C|t-s|^{-\frac32}\,\|f\|_1 \text{\ \ for all times $t,s$.}$$
For the case of time independent potentials V(x), (0.2) remains true if
$$\int_{\mathbb{R}^6} \frac{|V(x)|\;|V(y)|}{|x-y|^2} \, dxdy <(4\pi)^2\text{\ \ \ and\ \ \ }\|V\|_{\mathcal{K}}:=\sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|V(y)|}{|x-y|}\,dy<4\pi.$$
We also establish the dispersive estimate with an ε-loss for large energies provided \(\|V\|_{\mathcal{K}}+\|V\|_2<\infty\).

Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x|-2-ε in dimensions n≥3, thus solving an open problem posed by Journé, Soffer, and Sogge.