On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum
arXiv:0710.2331 · doi:10.1007/s10955-007-9419-5
Abstract
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*|m-n|^{-p}max{m,n}^{-2γ} for m!=n 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 εis small enough. More precisely, for any initial condition Ψ\in Dom(H^{1/2}), the diffusion of energy is bounded from above as <H>_Ψ(t)=O(t^σ) where σ=α/(2\ceil{p-1}γ-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.