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Note on a Product Formula Related to Quantum Zeno Dynamics

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Abstract

Given a nonnegative self-adjoint operator H acting on a separable Hilbert space and an orthogonal projection P such that \(H_P := (H^{1/2}P)^*(H^{1/2}P)\) is densely defined, we prove that \(\lim _{n\rightarrow \infty } (P\,\mathrm {e}^{-itH/n}P)^n = \mathrm {e}^{-itH_P}P\) holds in the strong operator topology. We also derive modifications of this product formula and its extension to the situation when P is replaced by a strongly continuous projection-valued function satisfying \(P(0)=P\).

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Acknowledgements

The research was supported by the Czech Science Foundation within the Project 17-01706S and in part by Grant-in Aid for Scientific Research 16K05230, Japan Society for the Promotion of Science, and by the EU Project CZ.02.1.01/0.0/0.0/16_019/0000778. The authors are grateful to Tsuyoshi Ando for valuable discussions, to Hiroshi Tamura, Valentin Zagrebnov, and late Hagen Neidhardt for a number of useful comments, and to Hideo Tamura for his unceasing encouragement.

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Authors and Affiliations

  1. Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519, Prague, Czechia

    Pavel Exner

  2. Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068, Řež near Prague, Czechia

    Pavel Exner

  3. Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa, 920-1192, Japan

    Takashi Ichinose

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  1. Pavel Exner
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  2. Takashi Ichinose
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Correspondence to Pavel Exner.

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Communicated by Alain Joye.

To the memory of our friend Hagen Neidhardt (1950–2019).

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Exner, P., Ichinose, T. Note on a Product Formula Related to Quantum Zeno Dynamics. Ann. Henri Poincaré 22, 1669–1697 (2021). https://doi.org/10.1007/s00023-020-01014-z

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