Blanchard, Ph.; Ma, Zhiming New results on the Schrödinger semigroups with potentials given by signed smooth measures. (English) Zbl 0717.60075 Stochastic analysis and related topics II, Proc. 2nd Workshop, Silivri/Turk. 1988, Lect. Notes Math. 1444, 213-243 (1990). [For the entire collection see Zbl 0706.00020.] The authors study the Schrödinger operator \(H^{\mu}=(\Delta /2-\mu)\) where \(\mu\) is given by a signed smooth measure and may be nowhere Radon in the sense that \(\mu (G)=\infty\) for all open sets G. They obtain an estimate of the integral kernel \(p^{\mu}(t,x,y)\) for the Schrödinger semigroup \((e^{tH^{\mu}})_{t>0}\). It is shown that under the compatibility condition \(p^{\mu}(t,x,y)\leq ce^{\beta z}p(\alpha t,x,y)\) for some constants \(c>0\), \(\alpha >0\), and \(-\infty <\beta <\infty\), where p(t,x,y) is the classical heat kernel. They obtain also a gauge theorem which generalizes the corresponding gauge theorems in their previous work [Smooth measures and Schrödinger semigroups, BiBoS preprint 295 (Bielefeld, 1987)]. Reviewer: Ph.Blanchard Cited in 13 Documents MSC: 60H25 Random operators and equations (aspects of stochastic analysis) Keywords:Schrödinger operator; Schrödinger semigroup; heat kernel; gauge theorems Citations:Zbl 0706.00020 PDFBibTeX XML