Note on a Product Formula Related to Quantum Zeno Dynamics

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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Correspondence to Pavel Exner.

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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 (2021). https://doi.org/10.1007/s00023-020-01014-z

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