Estimate of the difference between the Kac operator and the Schrödinger semigroup. (English) Zbl 0912.47025

Ichinose, Takashi; Takanobu, Satoshi

Commun. Math. Phys. 186, No. 1, 167-197 (1997).

The authors establish an \(L^p\)-operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup. This estimate is used to give a variant of the Trotter product formula for the Schrödinger operator \((1/2)\Delta+V\) in the \(L^p\)-operator norm. This extends Helffer’s result in the \(L^2\)-operator norm to the case of the \(L^p\)-operator norm for more general scalar potentials and the magnetic Schrödinger operator \([-i\partial-A(x)]^2/2\) with vector potential \(A(x)\) including the case of constant magnetic fields. The method of proof is probabilistic and is based on the Feynman–Kac and Feynman-Kac-Itô formulae.

Reviewer: R.Vaillancourt (Ottawa)

MSC:

47F05	General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35J10	Schrödinger operator, Schrödinger equation
47D06	One-parameter semigroups and linear evolution equations

Keywords:

Kac operator; Schrödinger semigroup; Trotter product formula; Feynman-Kac-Itô formula; \(L^p\)-operator norm estimate

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