Abstract
Let the pair of operators, (H, T), satisfy the weak Weyl relation:
where H is self-adjoint and T is closed symmetric. Suppose that g is a real-valued Lebesgue measurable function on \({\mathbb {R}}\) such that \({g\in C^2(\mathbb {R}\backslash K)}\) for some closed subset \({K\subset\mathbb {R}}\) with Lebesgue measure zero. Then we can construct a closed symmetric operator D such that (g(H), D) also obeys the weak Weyl relation.
Similar content being viewed by others
References
Arai A.: Generalized weak Weyl relation and decay of quantum dynamics. Rev. Math. Phys. 17, 1071–1109 (2005)
Arai, A.: On the uniqueness of weak Weyl representations of the canonical commutation relation. Lett. Math. Phys. (2008, in press)
Arai, A.: Necessary and sufficient conditions for a Hamiltonian with discrete eigenvalues to have time operators, mp-arc 08-154 (2008, preprint)
Arai A., Matsuzawa Y.: Construction of a Weyl representation from a weak Weyl representation of the canonical commutation relation. Lett. Math. Phys. 83, 201–211 (2008)
Arai, A., Matsuzawa, Y.: Time operators of a Hamiltonian with purely discrete spectrum. Rev. Math. Phys. (2008, in press)
Galapon E.A.: Self-adjoint time operator is the rule for discrete semi-bounded Hamiltonians. Proc. R. Soc. Lond. A 458, 2671–2689 (2002)
Galapon E.A., Caballar R.F., Bahague R.T. Jr: Confined quantum time of arrivals. Phys. Rev. Lett. 93, 180406 (2004)
Dorfmeister G., Dorfmeister J.: Classification of certain pairs of operators (P, Q) satisfying [P, Q] = − iId. J. Funct. Anal. 57, 301–328 (1984)
Fujiwara I.: Rational construction and physical signification of the quantum time operator. Prog. Theor. Phys. 64, 18–27 (1980)
Fujiwara I., Wakita K., Yoro H.: Explicit construction of time-energy uncertainty relationship in quantum mechanics. Prog. Theor. Phys. 64, 363–379 (1980)
Goto T., Yamaguchi K., Sudo N.: On the time opertor in quantum mechanics. Prog. Theor. Phys. 66, 1525–1538 (1981)
Goto T., Yamaguchi K., Sudo N.: On the time opertor in quantum mechanics. II. Prog. Theor. Phys. 66, 1915–1925 (1981)
Kobe D.H., Aguilera-Navarro V.C.: Derivation of the energy-time uncertainty relation. Phys. Rev. A 50, 933–938 (1994)
Lewis H.R., Laurence W.E., Harris J.D.: Quantum action-angle variables for the harmonic oscillator. Phys. Rev. Lett. 26, 5157–5159 (1996)
Miyamoto M.: A generalised Weyl relation approach to the time operator and its connection to the survival probability. J. Math. Phys. 42, 1038–1052 (2001)
Rosenbaum D.M.: Super Hilbert space and the quamntum-mechanical time operators. J. Math. Phys. 19, 1127–1144 (1969)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hiroshima, F., Kuribayashi, S. & Matsuzawa, Y. Strong Time Operators Associated with Generalized Hamiltonians. Lett Math Phys 87, 115–123 (2009). https://doi.org/10.1007/s11005-008-0287-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-008-0287-y