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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

Citations:

Zbl 0707.00017