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 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 σ = α/(2⌈p − 1⌉γ − 1 2 ). As an application we consider the Hamiltonian H(t) = |p| α + εv(θ, t) on L 2 (S 1 , dθ) which was discussed earlier in the literature by Howland.
Let H ( ) = − 2 d 2 /dx 2 + V (x) be a Schrödinger operator on the real line, W (x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V (x) in exactly two turning points and lies below V ∞ = lim inf |x|→∞ V (x). We consider the semiclassical limit n → ∞, = n → 0 and E n = E where E n is the nth eigenenergy of H ( ). An asymptotic formula for n|W (x)|n + k , the non-diagonal matrix elements of W (x) in the eigenbasis of H ( ), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.