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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

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)

Citations:

Zbl 0706.00020