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
Abstract
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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