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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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##### References:
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