Some remarks on the Cauchy problem for Schrödinger type equations (Q1062202)

The author considers the Cauchy problem for the equation \[ (*)\quad (i\partial_ t+\tau \Delta +\sum^{m}_{j=1}b_ j(x)\partial_{x_ j}+c(x))u(x,t)=0 \] with initial data \(u(x,0)=u(x_ 0)\). Here \(\tau\), \(0\leq \tau \leq 1\) is a constant and \(b_ j\), c are elements of \(C^{\infty}({\mathbb{R}}^ n_ x)\) and it is assumed that the functions \(b_ j\) and c together with all their derivatives are bounded. Note that for \(\tau >0\) the equation is of non-Kowalewskian type. The author proves the following theorem: In order that equation (*) is well posed for the future or for the past in the space \(H_{\infty}(:=\cap_{s\in {\mathbb{R}}}H_ s)\) for a given \(u_ 0\in H_{\infty}\), it is necessary that there exist constant M and N such that the inequality \[ \sup_{x\in {\mathbb{R}}^ m,w\in S^{m-1}}| \sum^{m}_{j=1}\int^{\rho}_{0}Re b_ j(x+2\tau \theta \omega)\omega_ j d\theta | \leq u \log (1+\rho)+N \] holds for any \(\rho\geq 0\). here \(S^{m-1}\) is the unit sphere in \({\mathbb{R}}^ m\).