Abstract
Stability properties of Feynman's operational calculus are addressed in the setting of exponential functions of noncommuting operators. Applications of some of the stability results are presented. In particular, the time-dependent perturbation theory of nonrelativistic quantum mechanics is presented in the setting of the operational calculus and application of the stability results of this paper to the perturbation theory are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blank, J., Exner, P. and Havlicek, M.: Hilbert Space Operators in Quantum Physics, AIP Press, New York, 1994.
Billingsley, P.: Convergence of Probability Measures, Wiley, New York, 1968.
Badrikian, A., Johnson, G. W. and Yoo, Il.: The composition of operator-valued measurable functions is measurable, Proc. Amer. Math. Soc. 123 (1995), 1815-1820.
Cohen-Tannoudji, C., Diu, B. and LaloË, F.: Quantum Mechanics (Engl. transl. of the 1973 French ed.), Vols. I and II, Herman and Wiley, Paris and New York, 1977.
Defacio, B., Johnson, G.W. and Lapidus, M. L.: Feynman's operational calculus and evolution equations, Acta Appl. Math. 47 (1997), 155-211.
Feynman, R. P.: An operator calculus having applications in quantum electrodynmanics, Phys. Rev. 84 (1951), 108-128.
Goldstein, J. A.: Semigroups of Linear Operators and Applications, Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 1985.
Hille, E. and Phillips, R. S.: Functional Analysis and Semi-Groups, rev. edn, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957.
Jefferies, B. and Johnson, G. W.: Feynman's operational calculi for noncommuting operators: Definitions and elementary properties, Russ. J. Math. Phys. 8 (2001), to appear.
Jefferies, B. and Johnson, G. W.: Feynman's operational calculi for noncommuting operators: Tensors, ordered supports and disentangling an exponential factor, Math. Notes 70 (2001), accepted for publication.
Jefferies, B., Johnson, G.W. and Nielsen, L.: Feynman's operational calculi for time-dependent noncommuting operators, J. Korean Math. Soc. 38 (2001), 193-226.
Johnson, G. W. and Lapidus, M. L.: The Feynman Integral and Feynman's Operational Calculus, Oxford Science Publications, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2000.
Johnson, G. W. and Nielsen, L.: A stability theorem for Feynman's operational calculus, In: F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner and S. Scarlatti (eds), Stochastic Processes, Physics and Geometry: New Interplays. II: A Volume in Honor of Sergio Albeverio, CMS Conf. Proc. 29, Amer. Math. Soc., Providence, 2000.
Nielsen, L.: Effects of absolute continuity in Feynman's operational calculus, Proc. Amer.Math. Soc., accepted for publication.
Nielsen, L.: Stability properties for Feynman's operational calculus, PhD Dissertation, Mathematics, University of Nebraska, Lincoln, 1999.
Nielsen, L.: Time dependent stability for Feynman's operational calculus, Submitted for publication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nielsen, L. Stability Properties of Feynman's Operational Calculus for Exponential Functions of Noncommuting Operators. Acta Applicandae Mathematicae 74, 265–292 (2002). https://doi.org/10.1023/A:1021155700823
Issue Date:
DOI: https://doi.org/10.1023/A:1021155700823