Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H 0(t) + W(t), where H 0(t) commutes for all twith a complete set of time-independent projectors {Pj}∞j=1. Consider the observable A=∑j P jλjwhere λ j ≃ j μ, μ>0, for jlarge. Assuming that the “matrix elements” of W(t) behave as for p>0 large enough, we prove estimates on the expectation value ⟨U(t)ϕ|AU(t)ϕ⟩≡⟨A⟩ϕ(t)for large times of the type where δ>0 depends on pand μ. Typical applications concern the energy expectation 〈H0〉ϕ(t) in case H 0(t) ≡ H 0or the expectation of the position operator 〈x2〉ϕ(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H 0(t) is a time-dependent multiplicative potential.