Integral Equations and Operator Theory

, Volume 69, Issue 4, pp 451–478 | Cite as

Remarks on the Trotter–Kato Product Formula for Unitary Groups

  • Pavel Exner
  • Hagen Neidhardt
  • Valentin A. Zagrebnov
Article

Abstract

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L2-norm, that is, one has
$$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $$
for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: \({\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}\), where \({{\mathbb R}_+ := [0,\infty)}\), satisfying in addition \({{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}\) and \({{\Im{\rm m}}\,(\psi(y)) \le 0}\) for \({y \in {\mathbb R}_+}\), we prove that
$$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $$
holds true uniformly on \({[0,T]\ni t}\) for any T > 0.

Mathematics Subject Classification (2010)

Primary 47A55 47D03 81Q30 Secondary 47B25 

Keywords

Trotter product formula Trotter–Kato product formula unitary groups Feynman path integrals holomorphic Kato functions admissible functions 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Pavel Exner
    • 1
    • 2
  • Hagen Neidhardt
    • 3
  • Valentin A. Zagrebnov
    • 4
  1. 1.Department of Theoretical Physics, NPIAcademy of SciencesŘežCzech Republic
  2. 2.Doppler Institute, Czech Technical UniversityPragueCzech Republic
  3. 3.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  4. 4.Centre de Physique Théorique, Université de la Méditerranée (Aix-Marseille II)Marseille Cedex 9France

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