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The norm estimate of the difference between the Kac operator and Schrödinger semigroup. II: The general case including the relativistic case. (English) Zbl 0987.47032
The authors’ previous paper [Commun. Math. Phys. 186, No. 1, 167-197 (1997; Zbl 0912.47025) and Nagoya Math. J. 149, 53-81 (1998; Zbl 0917.47041)] discussed the \(L_p\)-operator norm estimate of difference between the Kac operator and the Schrödinger semigroup. In the present paper, more general Schrödinger operators associate with the Levy processes, including the reletivistic Schrödinger operators, are studied and the results are generalised. As an application, the Trotter product formula in the \(L_p\)-operator norm is derived. The method of proof is based on using the Feynman-Kac formula not directly for the operator, but instead through subordination from the Brownian motion to enable to deal with all operators in a unified way.

MSC:
47D07 Markov semigroups and applications to diffusion processes
35J10 Schrödinger operator, Schrödinger equation
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
60J65 Brownian motion
60J35 Transition functions, generators and resolvents
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