Journal of Statistical Physics

, Volume 130, Issue 1, pp 169–193 | Cite as

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

  • Pierre Duclos
  • Ondra Lev
  • Pavel Šťovíček


We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E n n α , with 0<α<1. In particular, the gaps between successive eigenvalues decay as n α−1. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate ‖V(t) m,n ‖≤ε|mn|p max {m,n}−2γ for mn, where ε>0, p≥1 and γ=(1−α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and ε is small enough. More precisely, for any initial condition Ψ∈Dom(H 1/2), the diffusion of energy is bounded from above as 〈H Ψ (t)=O(t σ ), where \(\sigma=\alpha/(2\lceil p-1\rceil \gamma-\frac{1}{2})\) . As an application we consider the Hamiltonian H(t)=|p| α +ε v(θ,t) on L 2(S 1,dθ) which was discussed earlier in the literature by Howland.


Energy growth Periodically driven quantum system 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Centre de Physique Théorique de Marseille UMR 6207—Unité Mixte de Recherche du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’ Université du Sud Toulon-Var—Laboratoire affilié à la FRUMAMMarseille Cedex 9France
  2. 2.Department of Mathematics, Faculty of Nuclear ScienceCzech Technical UniversityPragueCzech Republic

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