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

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|^−pmax {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 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.