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Linear Adiabatic Theory: Exponential Estimates and Applications

  • G. Nenciu
Chapter
Part of the Mathematical Physics Studies book series (MPST, volume 19)

Abstract

The evolution equation
in the limit ε → 0, is considered. Various methods to construct asymptotically invariant (up to exponentially small errors) subspaces are reviewed. Then, following the basic idea of the reduction theory, the so called “superadiabatic evolution” is written down. In the second part some applications of the general theory are presented: theory of adiabatic invariants for linear Hamiltonian systems and spectral properties of periodic Dirac hamiltonian.

Keywords

Invariant Subspace Quantum Hall Effect Spectral Projection Spectral Concentration Exponential Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Avron, J.E., Seiler, R., and Yaffe, L.G.: Adiabatic theorem and applications to the quantum Hall effect, Commun. Math. Phys. 110 (1987) 33 – 49.MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Born, M. and Fock, V.: Beweis des Adiabatensatzes, Z. Phys. 5 (1928) 165 – 180.ADSCrossRefGoogle Scholar
  3. 3.
    Boutet de Monvel, A. and Nenciu, G.: On the theory of adiabatic invariants for linear Hamiltonian systems, C. R. Acad. Sci. Paris 310 (1990) 807 – 810.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berry, M.V.: Quantum phase corrections from adiabatic iteration, Proc. R. Soc. Lond. A414 (1987) 31 – 46.ADSCrossRefGoogle Scholar
  5. 5.
    Berry, M.V.: Histories of adiabatic quantum transitions, Proc. R. Soc. Lond. A429, 61–72(1990).Google Scholar
  6. 6.
    Garrido, L.M.: Generalized adiabatic invariance, J. Math. Phys. 5, (1964) 335 – 362.MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Jdanova, G. V. and Fedorjuk, M. V.: Asymptotic theory for the systems of second order differential equations and the scattering problem. Trudy. Mosk. Mat. Ob. 34 (1977) 213 – 242.Google Scholar
  8. 8.
    Joye, A.: Absence of absolutely continuous spectrum of Floquet operators, Preprint CNRS-CPT-93: P.2957, Marseille.Google Scholar
  9. 9.
    Joye, A. and Pfister, C-E.: Exponentially small adiabatic invariant for the Schrödinger equation, Commun. Math. Phys. 140 (1991) 15 – 41.MathSciNetADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Joye, A. and Pfister, C-E.: Full asymptotic expansion of transition probabilities in the adiabatic limit, J. Phys. A: Math. Gen. 24 (1991) 753 – 766.MathSciNetADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Joye, A. and Pfister, C-E.: Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum, J. Math. Phys. 34 (1993) 454 – 479.MathSciNetADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Joye, A. and Pfister, C-E.: Semi-classical asymptotics beyond all orders for simple scattering systems, Preprint, CNRS Luminy, Marseille, 1993.Google Scholar
  13. 13.
    Joye, A. and Pfister, C-E.: Quantum adiabatic evolution, In the Proceedings of the NATO Advanced Workshop “On three levels”, Ed. A. Verbure, Plenum Press 1994.Google Scholar
  14. 14.
    Kato, T.: On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan 5 (1950) 435 – 439.ADSCrossRefGoogle Scholar
  15. 15.
    Kato, T.: Perturbation theory for linear operators, Springer, Berlin, Heidelberg, New York 1976.Google Scholar
  16. 16.
    Krein, S. G.: Linear differential equations in Banach spaces, Translations of Mathematical Monographs, 29, Providence 1971.Google Scholar
  17. 17.
    Lenard, A.: Adiabatic invariance to all orders, Ann. Phys. 6 (1959) 261 – 276.MathSciNetADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Leung, A. and Meyer, K.: Adiabatic invariants for linear systems, J. Diff. Eq. 17 (1975) 32 – 43.MathSciNetCrossRefzbMATHADSGoogle Scholar
  19. 19.
    Levy, M.: Adiabatic invariants of linear Hamiltonian systems with periodic coefficients, J. Diff. Eq. 42 (1981) 47 – 71.CrossRefADSGoogle Scholar
  20. 20.
    Martinez, A.: Precise exponential estimates in adiabatic theory, Preprint Université Paris Nord 1993.Google Scholar
  21. 21.
    Martinez, A. and Nenciu, G.: In preparation.Google Scholar
  22. 22.
    Martin, Ph. A. and Nenciu, G.: Semi-classical inelastic S-matrix for one-dimensional N-states systems, Rev. Math. Phys. 7 (1995) 193 – 242.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nenciu, G.: On the adiabatic theorem of quantum mechanics, J. Phys. A: Math. Gen. 13 (1980) L15 – L18.MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Nenciu, G.: Adiabatic theorem and spectral concentration, Commun. Math. Phys. 82 (1981) 125 – 135.MathSciNetADSGoogle Scholar
  25. 25.
    Nenciu, G.:Adiabatic theorem and spectral concentration II. Arbitrary order asymptotic invariant subspaces and block diagonalisation, Preprint FT-308-1987, Central Institute of Physics, Bucharest.Google Scholar
  26. 26.
    Nenciu, G.: Asymptotic invariant subspaces, adiabatic theorems and block diagonalisation. In: Boutet de Monvel et al. (eds), Recent developments in quantum mechanics, 133 – 149, Kluwer, Dordrecht 1991.CrossRefGoogle Scholar
  27. 27.
    Nenciu, G.: Dynamics of band electrons in electric and magnetic fields: rigorous justi- fication of the effective Hamiltonians, Rev. Mod. Phys. 63 (1991) 91 – 128.ADSCrossRefGoogle Scholar
  28. 28.
    Nenciu, G.: Linear adiabatic theory. Exponential estimates, Commun. Math. Phys. 152 (1993) 479 – 496.MathSciNetADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Nenciu, G.: Floquet operators without absolutely continuous spectrum, Ann. Inst. Henri Poincaré: Phys. Theor. 59 (1993) 91 – 97.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Nenciu, G. and Purice, R.: One dimensional periodic Dirac hamiltonians: semiclassical and high-energy asymptotics for gaps, J. Math. Phys. (to appear).Google Scholar
  31. 31.
    Nenciu, G. and Rasche, G.: Adiabatic theorem and Gell-Mann-Low formula, H. P. A. 62 (1989) 372 – 388.MathSciNetGoogle Scholar
  32. 32.
    Wasow, W.: Topics in the theory of linear differential equations having singularities with respect to a parameter, Série de Mathématiques Pures et Apliquées, IRMA, Strasbourg, 1978.Google Scholar
  33. 33.
    Wostarowski, M. P.: A remark on strong stability of linear Hamiltonian systems, J. Diff. Eq. 81 (1989) 313 – 316.CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • G. Nenciu
    • 1
    • 2
  1. 1.Department Theoretical PhysicsUniversity of BucharestBucharestRomania
  2. 2.Institute of Mathematics of Romanian AcademyBucharestRomania

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