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Pré-Publication, Document De Travail Année : 2008

On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum

Résumé

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^\alpha, with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay as n^{\alpha-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}|<=\epsilon*|m-n|^{-p}max{m,n}^{-2\gamma} for m!=n where \epsilon>0, p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and \epsilon is small enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where \sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was discussed earlier in the literature by Howland.
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Dates et versions

hal-00184770 , version 1 (11-01-2024)

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Pierre Duclos, Ondra Lev, Pavel Stovicek. On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum. 2007. ⟨hal-00184770⟩
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