## On smoothing property of Schrödinger propagators.(English)Zbl 0725.35084

Functional-analytic methods for partial differential equations, Proc. Conf. Symp., Tokyo/Jap. 1989, Lect. Notes Math. 1450, 20-35 (1990).
[For the entire collection see Zbl 0707.00017.]
The smoothing property of the propagator is discussed for time dependent Schrödinger equations with real potentials A(t,x) and V(t,x) in $$R^ n:$$ $(E)\quad i\partial_ tu=(-i\partial_ x-A(t,x))^ 2u+V(t,x)u.$ The assumptions are
(A.1) $$\partial_ x^{\alpha}A(t,x)$$ is $$C^ 1$$. If $$| \alpha | \geq 1$$, $$| \partial_ x^{\alpha}rot A(t,x)| \leq C_{\alpha}(1+| x|)^{-1-\epsilon}$$, $$\epsilon >0$$, and $$| \partial_ x^{\alpha}A(t,x)| +| \partial_ x^{\alpha}\partial_ tA(t,x)| \leq C;$$
(A.2) $$\partial_ x^{\alpha}V(t,x)$$ is continuous and if $$| \alpha | \geq 2$$, $$| \partial_ x^{\alpha}V(t,x)| \leq C_{\alpha}.$$
Under (A.1) and (A.2), (E) generates a unique unitary propagator U(t,s) in $$L^ 2(R^ n)$$. The main results are the following. $$T>0$$ is a sufficiently small number.
Theorem 1. Let $$\mu >1/2$$ and $$\rho\geq 0$$. Then $\| <x>^{-\mu - \rho}<D>^{\rho}U(\cdot,0)f\|_{L^ 2([-T,T],L^ 2(R^ n))}\leq C\| <D>^{\rho -1/2}f\|_{L^ 2(R^ n)}.$ Theorem 2.Let $$p\geq 2$$, $$0\leq 2/\theta =2\sigma +n(1/2-1/p)<1$$ and $$\rho\in R$$. Then, $\| <x>^{-2\sigma -| \rho |}<D>^{\rho +\sigma}U(\cdot,0)f\|_{L^{\theta}([-T,T],L^ p(R^ n))}\leq C\| <D>^{\rho}f\|_{L^ 2(R^ n)}.$ Theorem 3. Let $$\gamma >1/2$$ and $$\delta >2\gamma +1/2$$. Then $\| \sup_{| t| \leq T}| <x>^{-\delta}U(t,0)f(x)| \|_{L^ 2(R^ n)}\leq C\| <D>^{\gamma}f\|_{L^ 2({\mathbb{R}}^ n)}.$ Theorem 4. Let $$f\in H^{\gamma}(R^ n)\cap L^ 1(R^ n)$$ with $$\gamma >1/2$$. Then,
$$\lim_{t\to 0}U(t,0)f(x)=f(x)$$, a.e. $$x\in R^ n.$$
These are extensions of the known results of, say, Constantin-Saut or Sjolin for the free Schrödinger equation.
Reviewer: Kenji Yajima

### MSC:

 35Q40 PDEs in connection with quantum mechanics 35B65 Smoothness and regularity of solutions to PDEs 35B45 A priori estimates in context of PDEs

### Keywords:

smoothing property; propagator; Schrödinger equations

Zbl 0707.00017