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We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrumof 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 decayas n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuouslydifferentiable in the strong sense and such that the matrix entries withrespect 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 bearbitrarily small provided p is sufficiently large and \epsilon is smallenough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), thediffusion of energy is bounded from above as <H>_\Psi(t)=O(t^\sigma) where\sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider theHamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which wasdiscussed earlier in the literature by Howland.

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