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

Abstract

In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions

$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$

(0.1)

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 f∈L ²(ℝ³) satisfying the dispersive inequality

$$\|\psi(t)\|_{\infty} \le C|t-s|^{-\frac32}\,\|f\|_1 \text{\ \ for all times $t,s$.}$$

(0.2)

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.