- Access by Xinjiang University
Method of Averaging and the Quantum Anharmonic Oscillator
Phys. Rev. Lett. 55, 1950 – Published 4 November, 1985
DOI: https://doi.org/10.1103/PhysRevLett.55.1950
Abstract
The Krylov-Bogoliubov method of averaging is applied to the time-dependent quantum anharmonic oscillator. A regular perturbation expansion contains secular terms. The averaging approximation does not, and as a result has a validity over larger time intervals. A new variant of the usual averaging transformation is used and rigorous error bounds are derived. Rigorous averaging methods have been applied extensively to ordinary differential equations but our work appears to be the first generalization to partial differential equations.
References (12)
- N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1961), Chaps. 5 and 6 J. K. Hale, Ordinary Differential Equations (Wiley-Interscience, New York, 1969), 1st ed., Chap. 5 V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, 1983), Chap. 4 T. J. Burns and J. A. Ellison, Phys. Rev. B 29, 2790 (1984) H. S. Dumas and J. A. Ellison, "Particle Channeling in Crystals and the Method of Averaging," in Proceedings of the Workshop on Local and Global Methods of Dynamics, edited by R. Cawley, A. W. Sáenz, and W. W. Zachary (Springer, New York, to be published) H. S. Dumas, J. A. Ellison, and A. W. Sáenz, to be published C. Robinson, IEEE Trans. Circuits Syst. 30, 591 (1983)
- M. Kummer, Nuovo Cimento B1, 123 (1971)
- Burns and Ellison, Dumas and Ellison, and Dumas, Ellison, and Sáenz, [1]
- M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, New York, 1972)
- M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic, New York, 1972), p. 265, Theorem VIII.7
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977), p. 136
- I. D. Feranchuk and L. I. Komarov, Phys. Lett. 88A, 211 (1982)
- Hale, [1]
- Robinson, [1]
- A. Ben Lemlih, thesis, University of New Mexico (to be published)
- R. J. Cook, D. G. Shankland, and A. L. Wells, Phys. Rev. A 31, 564 (1985)
- E. A. Coutsias and J. K. McIver, Phys. Rev. A 31, 3155 (1985)