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Adiabatic theory, Liapunov exponents, and rotation number for quadratic Hamiltonians. (English) Zbl 0965.37506

Summary: 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 Lyapunov 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.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70H05 Hamilton’s equations
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References:

[1] M. V. Fedoryuk,Equations Differentielles 12:6 (1976). · Zbl 0449.58002
[2] A. I. Neishtadt,Prikl Matem. Mekhan. 45:80 (1981).
[3] F. Delyon and P. Foulon,J. Stat. Phys. 45:41 (1986). · Zbl 0628.34052 · doi:10.1007/BF01033075
[4] T. Kato,Phys. Soc. Jpn. 5 (1950).
[5] M. Berry,Proc. R. Soc. A 392:45 (1984). · Zbl 1113.81306 · doi:10.1098/rspa.1984.0023
[6] F. V. Atkinson,Discrete and Continuous Boundary Problems (Academic Press, 1964). · Zbl 0117.05806
[7] D. Ruelle,Ann. Inst. H. Poincaré Phys. Théor. 42:109 (1985).
[8] F. Delyon and P. Foulon, Complex entropy for dynamical systems, Preprint, Ecole Polytechnique. · Zbl 0748.58009
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