The \(L_ 2\)-well-posed Cauchy problem for Schrödinger type equations
The author treats the following Cauchy problem for equations of Schrödinger type: \[ \begin{cases} {\partial u \over i \partial t}+{1 \over 2}\sum^ n_{j=1} \left( {\partial u \over i \partial x_ j}-a_ j(x,t) \right)^ 2+c(x,t)=0 & \text{ in } \{0<t<T,\;x \in \mathbb{R}^ n\} \\ u(0,x)=u_ 0(x) & \text{ on } x \in \mathbb{R}^ n \end{cases} , \tag{*} \] and he looks for a sufficient condition so that the Cauchy problem \((*)\) is well-posed in \(L^ 2\)-space. Let us write \(a_ j(x,t) =a^ R_ j(x,t)+ia^ I_ j(x,t)\) where \(a^ R_ j(x,t)\) and \(a^ I_ j(x,t)\) are real valued functions. \textit{J. Takeuchi} [C. R. Acad. Sci., Paris, Sér. I 310, 823-826 (1990; Zbl 0721.35024)] proved that, if there exists a constant \(\varepsilon>0\) such that \(| \partial^ \alpha_ xa^ I_ j(x) | \leq C_ \alpha \langle x \rangle^{-1-\varepsilon-| \alpha |}\) for all \(\alpha\) and \(| \partial^ \alpha_ xa^ R_ j(x)| \leq C_ \alpha \langle x \rangle^{-1}\) for \(| \alpha | \geq 1\) \((j=1,2,\ldots,n)\) where \(\langle x \rangle=\sqrt {1+| x |^ 2}\) and each \(C_ \alpha\) is a constant which depends only on \(\alpha\), then the Cauchy problem \((*)\) is well-posed in \(L^ 2\)-space. In this paper, the author improves the above result as follows: If there exist \(\varepsilon_ 1\), \(\varepsilon_ 2>0\) such that \[ \bigl| a^ I_ j(x,t)\bigr| \leq C_ 0 \langle x \rangle^{-1- \varepsilon_ 1},\;\bigl| \partial^ \alpha_ xa^ I_ j(x,t) \bigr| \leq C_ \alpha \langle x \rangle^{-2-\varepsilon_ 1} \] for \(| \alpha | \geq 1\) and \(\bigl| \partial^ \alpha_ xa^ R_ j(x,t)\bigr| \leq C_ \alpha \langle x \rangle^{-1- \varepsilon_ 2}\) for \(| \alpha | \geq 2\) \((j=1,2,\ldots,n)\), then the Cauchy problem \((*)\) is well-posed in \(L^ 2\)-space.
scientific article; zbMATH DE number 109341