Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

Abstract

We consider the time-dependent Schrödinger-Hartree equation

$$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$

((1))

$$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$

((2))

where λ≧0 and$\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \}$.

We show that there exists a unique global solutionu of (1) and (2) such that

$$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$

with

$$u \in L^\infty (\mathbb{R};L^2 ).$$

Furthermore, we show thatu has the following estimates:

$$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$

and

$$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$