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

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‖≤ɛ|m‑n|^‑pmax {m,n}^‑2γ for m≠n, where ɛ>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 ɛ 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+ɛv(θ,t) on L²(S¹,dθ) which was discussed earlier in the literature by Howland.