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

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

2007-10-11

arxiv.org/abs/0710.2331v1

Physics

Mathematical Physics

Scientific paper

10.1007/s10955-007-9419-5

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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