Advertisement

Asymptotic Invariant Subspaces, Adiabatic Theorems and Block Diagonalisation

  • G. Nenciu
Chapter
Part of the Mathematical Physics Studies book series (MPST, volume 12)

Abstract

We consider the evolution iε \(frac{{\text{d}}}{{{\text{ds}}}}\)U(s) = H(s)U(s), U(0) = 1 fore ε→ 0. A recurrence procedure providing arbitrary order asymptotic invariant subspaces of U(s) corresponding to the isolated parts of the spectrum of H(s) is written down. The point of the construction is that at the kth step the invariant subspaces at the “time” s are constructed from H and its first k derivatives at the same time. As consequences we give a hierarchy of adiabatic theorems as well as a block diagonalisation scheme.

Keywords

Invariant Subspace Transformation Function Spectral Projection Adiabatic Theorem Linear Hamiltonian System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Born, M. Fock, V., Beweis des Adiabatensatzes, Z.Phys. 5, 165–180 (1928).ADSGoogle Scholar
  2. 2.
    Kato, T., On the adiabatic theorem of quantum mechanics, J.Phys.Soc.Japan 5, 435–439 (1950).ADSCrossRefGoogle Scholar
  3. 3.
    Garrido, L.M., Generalized adiabatic invariance, J.Math.Phys. 5, 335–362 (1964).MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Nenciu, G., Adiabatic theorem and spectral concentration, Commun.Math.Phys. 82, 121–135 (1981).MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Avron, J.E., Seiler, R., Yaffe, L.G., Adiabatic theorem and applications to the quantum Hall effect, Commun. Math.Phys. 110, 33–49 (1987).MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Krein, S.G., Linear differential equations in Banach spaces, A.M.S. Translations of Mathematical Monographs, vol.29, Providence, 1971.Google Scholar
  7. 7.
    Wasow, W., Topics in the theory of linear ordinary differential equations having singularities with respect to a parameter, Series de Mathématiques Pures et Appliquées, IRMA Strasbourg, 1978.Google Scholar
  8. 8.
    Jdanova, G.V., Fedoriuk, M.V., Asymptotic theory for the systems of second order differential equations and the scattering problem, Trudy Mosk.Mat.Ob. 34, 213–242 (1977).Google Scholar
  9. 8.
    Jdanova, G.V., Fedoriuk, M.V., Asymptotic theory for the systems of second order differential equations and the scattering problem, Trudy Mosk.Mat.Ob. 34, 213–242 (1977).Google Scholar
  10. 10.
    Nenciu, G., Rasche, G., Adiabatic theorem and Gell-Mann-Low formula, H.P.A. 62, 372–388 (1989).MathSciNetGoogle Scholar
  11. 11.
    Tanabe, H., Equations of Evolution, Pitman, Berlin, 1966.Google Scholar
  12. 12.
    Kato, T. Perturbation Theory for Linear Operators, Springer, Heidelberg, 1976.zbMATHGoogle Scholar
  13. 13.
    Reed, M., Simon, B., Methods of Modern Mathematical Physics, IV, Academic Press, New York, 1978.Google Scholar
  14. 14.
    Messiah, A., Quantum Mechanics, II, North Holland, Amsterdam, 1969.Google Scholar
  15. 15.
    Nenciu, A., Nenciu, G., Dynamics of Bloch Electrons in Electric Fields I, II, J.Phys. A14, 2817–2835 (1981); J.Phys. Als, 3313–3331 (1982).Google Scholar
  16. 16.
    Nenciu, G., Dynamics of band electrons in electric and magnetic fields: Rigorous justification of the effective Hamiltonians, Rev.Mod.Phys. to be published.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • G. Nenciu
    • 1
  1. 1.Theoretical Physics DepartmentInstitute of Atomic PhysicsBucharestRomania

Personalised recommendations