Note on a Product Formula Related to Quantum Zeno Dynamics


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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  1. 1.

    Beskow, A., Nilsson, J.: The concept of wave function and the irreducible representations of the Poincaré group, II. Unstable systems and the exponential decay law. Arkiv Fys. 34, 561–569 (1967)

    MATH  Google Scholar 

  2. 2.

    Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238–242 (1968)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chernoff, P.R.: Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators, Memoirs of the American Mathematical Society, vol. 140. R.I, AMS, Providence (1974)

  4. 4.

    Exner, P.: Open Quantum Systems and Feynman Integrals, Fundamental Theories of Physics, vol. 6. D. Reidel Publ. Co., Dordrecht (1985)

    Google Scholar 

  5. 5.

    Exner, P., Ichinose, T.: A product formula related to quantum Zeno dynamics. Ann. Henri Poincaré 6(2), 195–215 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Exner, P., Ichinose, T., Neidhardt, H., Zagrebnov, V.A.: Zeno product formula revisited. Integral Eq. Oper. Theory 57(1), 67–81 (2007)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Facchi, P., Pascazio, S.: Quantum Zeno dynamics: mathematical and physical aspects. J. Phys. A Math. Theor. 41(49), 493001 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Facchi, P., Pascazio, S., Scardicchio, A., Schulman, L.S.: Zeno dynamics yields ordinary constraints. Phys. Rev. A 65, 012108 (2001)

    ADS  Article  Google Scholar 

  9. 9.

    Friedman, C.N.: Semigroup product formulas, compressions and continuous observation in quantum mechanics. Indiana Univ. Math. J. 21, 1001–1011 (1972)

    Article  Google Scholar 

  10. 10.

    Glimm, J., Jaffe, A.: Singular perturbations of selfadjoint operators. Commun. Pure Appl. Math. 22, 401–414 (1969)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Halmos, P.: A Hilbert Space Problem Book. D. Van Nostrand Co., Inc., Princeton, N.J. (1967)

    Google Scholar 

  12. 12.

    Hodges, A.: What would Alan Turing have done after 1954? In: Teuscher, Ch. (ed.) Alan Turing: Life and Legacy of a Great Thinker, pp. 43–58. Springer, Heidelberg (2004)

    Google Scholar 

  13. 13.

    Ichinose, T.: The modified unitary Trotter-Kato and Zeno product formulas revisited. In: Dittrich, J., Kovařík, H., Laptev, A. (eds.) Functional Analysis and Operator Theory for Quantum Physics, pp. 401–417. EMS Publ, Zürich (2017)

    Google Scholar 

  14. 14.

    Itano, W.M., Heinzen, D.J., Bollinger, J.J., Wineland, D.J.: Quantum Zeno effect. Phys. Rev. A 41, 2295–2300 (1990)

    ADS  Article  Google Scholar 

  15. 15.

    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)

    Google Scholar 

  16. 16.

    Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. In: Topics in functional analysis (essays dedicated to M. G. Krein on the occasion of his 70th birthday), Adv. Math. Suppl. Stud., vol. 3, pp. 185–195; Academic Press, New York (1978)

  17. 17.

    Kelley, J.L., Namioka, I.: Linear topological spaces, with the collaboration of W.F. Donoghue, Jr., Kenneth R. Lucas, B.J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W.R. Scott, and Kennan T. Smith. Second corrected printing. Graduate Texts in Mathematics, vol. 36. Springer, New York (1976)

  18. 18.

    Köthe, G.: Topologische lineare Räume. I, Die Grundlehren der mathematischen Wissenschaften, Band 107. Springer, Berlin 1966; Topological vector spaces. I, Band 159, English translation by D.J.H. Garling (1969)

  19. 19.

    Matolcsi, M., Shvidkoy, R.: Trotter’s product formula for projections. Arch. der Math. 81, 309–317 (2003)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Misra, B., Sudarshan, E.C.G.: The Zeno’s paradox in quantum theory. J. Math. Phys. 18, 756–763 (1977)

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, Revised and Enlarged edn. Academic Press, New York, London (1980)

    Google Scholar 

  22. 22.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, New York, London (1978)

    Google Scholar 

  23. 23.

    Simon, B.: Operator Theory. A Comprehensive Course of Analysis, Part 4. AMS, Providence, R.I. (2015)

    Google Scholar 

  24. 24.

    Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)

    MathSciNet  Article  Google Scholar 

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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).

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