Summary: 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 \(E_{n}\sim n^{\alpha}\), 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
\[ \|V(t)_{m, n}\|\leq \varepsilon | m - n |^{- p}\max\{m, n\}^{- 2 \gamma} \quad \text{for }m \not= n, \] where \(\varepsilon >0, p \geq 1\) 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 \(\Psi \in \text{Dom}(H^{1/2})\), the diffusion of energy is bounded from above as \(\langle H \rangle_{\Psi}(t)= O (t^{\sigma})\), where \(\sigma=\alpha/(2\lceil p-1\rceil \gamma-\frac{1}{2})\). As an application we consider the Hamiltonian \(H (t)=| p |^{\alpha}+ \varepsilon v (\theta, t)\) on \(L^{2}(S^{1},d \theta)\) which was discussed earlier in the literature by Howland.