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

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
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