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Accretive perturbations and error estimates for the Trotter product formula

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Abstract

We study the operator-norm error bound estimate for the exponential Trotter product formula in the case of accretive perturbations. LetA be a semibounded from below self-adjoint operator in a separable Hilbert space. LetB be a closed maximal accretive operator such that, together withB *, they are Kato-small with respect toA with relative bounds less than one. We show that in this case the operator-norm error bound estimate for the exponential Trotter product formula is the same as for the self-adjointB [12]:

$$\parallel (e^{ - tA/n} e^{ - tB/n} )^n - e^{ - t(A + B)} \parallel \leqslant L\frac{{\ln n}}{n},n = 2,3,....$$

We verify that the operator—(A+B) generates a holomorphic contraction semigroup. One gets similar results whenB is substituted byB *.

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Authors and Affiliations

  1. Centre de Physique Théorique, CNRS-Luminy-Case 907, F-13288, Marseille Cedex 9, France

    Vincent Cachia & Valentin A. Zagrebnov

  2. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-10117, Berlin, Germany

    Hagen Neidhardt

  3. Départment de Physique, Université de la Méditerranée (Aix-Marseille II), F-13288, Marseille Cedex 9, France

    Valentin A. Zagrebnov

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  1. Vincent Cachia
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  2. Hagen Neidhardt
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  3. Valentin A. Zagrebnov
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To the memory of Tosio Kato

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Cachia, V., Neidhardt, H. & Zagrebnov, V.A. Accretive perturbations and error estimates for the Trotter product formula. Integr equ oper theory 39, 396–412 (2001). https://doi.org/10.1007/BF01203321

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