Communications in Mathematical Physics

, Volume 177, Issue 2, pp 327–347

Floquet Hamiltonians with pure point spectrum

  • P. Duclos
  • P. Šťovíček
Article

DOI: 10.1007/BF02101896

Cite this article as:
Duclos, P. & Šťovíček, P. Commun.Math. Phys. (1996) 177: 327. doi:10.1007/BF02101896

Abstract

We consider Floquet Hamiltonians of the type\(K_F : = - i\partial _t + H_0 + \beta V(\omega t)\), whereH0, a selfadjoint operator acting in a Hilbert space ℋ, has simple discrete spectrumE1<E2<... obeying a gap condition of the type inf {n−α(En+1−En); n=1, 2,...}>0 for a given α>0,t↦V(t) is 2π-periodic andr times strongly continuously differentiable as a bounded operator on ℋ, ω and β are real parameters and the periodic boundary condition is imposed in time. We show, roughly, that providedr is large enough, β small enough and ω non-resonant, then the spectrum ofKf is pure point. The method we use relies on a successive application of the adiabatic treatment due to Howland and the KAM-type iteration settled by Bellissard and extended by Combescure. Both tools are revisited, adjusted and at some points slightly simplified.

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • P. Duclos
    • 1
    • 2
  • P. Šťovíček
    • 3
  1. 1.Centre de Physique ThéoriqueCNRSMarseille-LuminyFrance
  2. 2.PHYMATUniversité de Toulon et du VarLa Garde CedexFrance
  3. 3.Department of Mathematics and Doppler Institute, Faculty of Nuclear ScienceCTUPragueCzech Republic