Communications in Mathematical Physics

, Volume 177, Issue 2, pp 327–347 | Cite as

Floquet Hamiltonians with pure point spectrum

  • P. Duclos
  • P. Šťovíček
Article

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.

Keywords

Boundary Condition Neural Network Statistical Physic Hilbert Space Complex System 

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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

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