The Norm Estimate of the Difference Between the Kac Operator and Schrödinger Semigroup II: The General Case Including the Relativistic Case

Takashi Ichinose (Kanazawa University)
Satoshi Takanobu (Kanazawa University)


More thorough results than in our previous paper in Nagoya Math. J. are given on the $L_p$-operator norm estimates for the Kac operator $e^{-tV/2} e^{-tH_0} e^{-tV/2}$ compared with the Schrödinger semigroup $e^{-t(H_0+V)}$. The Schrödinger operators $H_0+V$ to be treated in this paper are more general ones associated with the Lévy process, including the relativistic Schrödinger operator. The method of proof is probabilistic based on the Feynman-Kac formula. It differs from our previous work in the point of using the Feynman-Kac formula not directly for these operators, but instead through subordination from the Brownian motion, which enables us to deal with all these operators in a unified way. As an application of such estimates the Trotter product formula in the $L_p$-operator norm, with error bounds, for these Schrödinger semigroups is also derived.

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Pages: 1-47

Publication Date: January 26, 2000

DOI: 10.1214/EJP.v5-61


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