Advertisement

Almost Invariant Subspaces for Quantum Evolutions

  • G. Nenciu
Conference paper
Part of the Trends in Mathematics book series (TM)

Summary

In the first part the general framework of almost invariant subspaces for quantum evolutions (which can be viewed as a far-reaching generalization of the standard reduction theory [Ka] which lies at the core of Rellich—Kato theory of analytic perturbations) is reviewed. As examples, in the second part, (non-convergent) expansions leading to almost invariant subspaces are presented in more detail for time-dependent perturbations, as well as for the semi-classical limit.

Keywords

Invariant Subspace Secular Divergence Reduction Theory Quantum Evolution Heuristic Level 
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. [AES]
    M. Aizenman, A. Elgart, G.H. Schenker, Adiabatic charge transport and the Kubo formula for 2D Hall conductance, preprint.Google Scholar
  2. [ANe]
    A. Nenciu, Spectrul electronilor Bloch in prezenta campurilor externe slab neuniforme Studii si Cercetari de Fizica 39 (1987), 494–545.MathSciNetGoogle Scholar
  3. [ASY]
    J. Avron, R. Seiler, L.G. Yaffe, Adiabatic theorems and applications to the QHE Commun. Math. Phys. 110 (1987), 33–49.MathSciNetMATHCrossRefGoogle Scholar
  4. [AE]
    J. Avron, A. Elgart, Adiabatic theorem without a gap condition Corn- mun.Math. Phys. 203 (1999), 445–463.MathSciNetMATHGoogle Scholar
  5. [BN]
    R. Brummelhuis, J. Nourrigat, Scattering amplitude for Dirac operators Common. Partial Diff. Equations 24 (1999), 377–394.MathSciNetMATHCrossRefGoogle Scholar
  6. [ES]
    A. Elgart, G.H. Schenker, Adiabatic charge transport and the Kubo for-mula for Landau type hamiltonians, preprint.Google Scholar
  7. [EW]
    C. Emmerich, A. Weinstein, Geometry of transport equation in multicom-ponent WEB approximations Commun. Math. Phys. 176 (1996), 701–711.CrossRefGoogle Scholar
  8. [GMS]
    V. Grecchi, A. Martinez, A. Sacchetti, Destruction of the beating effect for a non-linear Schrödinger equation Commun. Math. Phys. 227 (2002), 191–209.MathSciNetMATHCrossRefGoogle Scholar
  9. [HS]
    B. Helffer, J. Sjöstrand, Analyse semiclassique pour l’équation de HarperII Mem. Soc. Math. France., Nouv. Serie. 40 (1990), 1–139.Google Scholar
  10. [JP1]
    A. Joye, Ch-E. Pfister, Superadiabatic evolution and adiabatic transition probability between two non-degenerate levels isolated in the spectrum J. Math. Phys. 34 (1993), 454–479.MathSciNetMATHCrossRefGoogle Scholar
  11. [JP2]
    A. Joye, Ch-E. Pfister, Exponential estimates in adiabatic quantum evolutions. In: D.E.Witt, A.J.Bracken, M.D.Gould, P.A.Pearce, eds. XIIth International Congress of Mathematical Physics (ICMP’97). Google Scholar
  12. [Ka]
    T. Kato Perturbation Theory for Linear Operators 2nd ed., Classics in Mathematics, Springer-Verlag, Berlin, 1980.MATHGoogle Scholar
  13. [LEK]
    P.W. Langhoff, S.T. Epstein, M. Karplus Rev. Mod. Phys. 44 (1972), 602.MathSciNetCrossRefGoogle Scholar
  14. [MN1]
    Ph-A. Martin, G. Nenciu, Perturbation theory for time dependent hamiltonians: rigorous reduction theory Heiv. Phys. Acta. 65 (1992), 528–559.MathSciNetGoogle Scholar
  15. [MN2]
    Ph-A. Martin, G. Nenciu, Semi-classical inelastic S-matrix for one-dimensional N-states systems Rev. Math. Phys. 7 (1995), 193–242.MathSciNetMATHCrossRefGoogle Scholar
  16. [Ma]
    A. Martinez, An Introduction to Semiclassical and Microlocal Analysis Springer, Berlin, 2001.Google Scholar
  17. [MaNE]
    A. Martinez, G. Nenciu, On adiabatic reduction theory Oper. Theory: Ad. Appl. 78 (1995), 243–252.MathSciNetGoogle Scholar
  18. [Ne0]
    G. Nenciu, Adiabatic theorem and spectral concentration Commun. Math. Phys. 82 (1981), 121–135.MathSciNetMATHCrossRefGoogle Scholar
  19. [Ne1]
    G. Nenciu, Asymptotic invariant subspaces, adiabatic theorems and block diagonalization, in Recent Developments in Quantum Mechanics, Boutet de Monvel et al., eds., Kluwer, Dordrecht, 1991.Google Scholar
  20. [Ne2]
    G. Nenciu, Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective hamiltonians Rev. Mod. Phys. 63 (1991), 91–128.CrossRefGoogle Scholar
  21. [Ne3]
    G. Nenciu, Linear adiabatic theory; exponential estimates Commun. Math. Phys. 152 (1993), 479–496.MathSciNetMATHCrossRefGoogle Scholar
  22. [Ne4]
    G. Nenciu, Linear adiabatic theory: Exponential estimates and applications, in Algebraic and Geometric Methods in Mathematical Physics, Boutet de Monvel and V. Marcenco, eds., Kluwer, Dordrecht, 1996.Google Scholar
  23. [Ne5]
    G. Nenciu, On asymptotic perturbation theory for quantum mechanics: almost invariant subspaces and gauge invariant magnetic perturbation theory J. Math. Phys. 43 (2002), 1273–1298.MathSciNetMATHCrossRefGoogle Scholar
  24. [Ne6]
    G. Nenciu, On asymptotic perturbation theory for quantum mechanics, in Long Time Behavior of Classical and Quantum Systems, S. Graffi and A. Martinez, eds.,World Scientific, Singapore, 2001.Google Scholar
  25. [NS]
    G. Nenciu, Vania Sordoni, Semi-classical limit for multistage Klein-Gordon systems: almost invariant subspaces and scattering theory. Preprint mp-arc 01–36 (2001). 7 Almost Invariant Subspaces for Quantum EvolutionsGoogle Scholar
  26. [Sj]
    J. Sjöstrand, Projecteurs adiabatiques du point de vue pseudodifferéntiel, C. R. Acad. Sci. Paris 317, Série I (1993), 217–220.Google Scholar
  27. [Te]
    S. Teufel Adiabatic Perturbation Theory in Quantum Dynamics Habilita-tionsschrift, Zentrum Mathematik, Technische Universität München, 2002.Google Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

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

Personalised recommendations