Abstract
We consider the adiabatic problem for general time-dependent quadratic Hamiltonians and develop a method quite different from WKB. In particular, we apply our results to the Schrödinger equation in a strip. We show that there exists a first regular step (avoiding resonance problems) providing one adiabatic invariant, bounds on the Liapunov exponents, and estimates on the rotation number at any order of the perturbation theory. The further step is shown to be equivalent to a quantum adiabatic problem, which, by the usual adiabatic techniques, provides the other possible adiabatic invariants. In the special case of the Schrödinger equation our method is simpler and more powerful than the WKB techniques.
References
M. V. Fedoryuk,Equations Differentielles 12:6 (1976).
A. I. Neishtadt,Prikl Matem. Mekhan. 45:80 (1981).
F. Delyon and P. Foulon,J. Stat. Phys. 45:41 (1986).
T. Kato,Phys. Soc. Jpn. 5 (1950).
M. Berry,Proc. R. Soc. A 392:45 (1984).
F. V. Atkinson,Discrete and Continuous Boundary Problems (Academic Press, 1964).
D. Ruelle,Ann. Inst. H. Poincaré Phys. Théor. 42:109 (1985).
F. Delyon and P. Foulon, Complex entropy for dynamical systems, Preprint, Ecole Polytechnique.
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Delyon, F., Foulon, P. Adiabatic theory, Liapunov exponents, and rotation number for quadratic Hamiltonians. J Stat Phys 49, 829–840 (1987). https://doi.org/10.1007/BF01009359
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DOI: https://doi.org/10.1007/BF01009359