Expectation Values of Observables in Time-Dependent Quantum Mechanics

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Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H0(t) + W(t), where H0(t) commutes for all t with a complete set of time-independent projectors {Pj}∞j = 1. Consider the observable A = ∑jPjλj, where λj ≃ jμ, μ>0, for j large. Assuming that the “matrix elements” of W(t) behave as ‖PjW(t)Pk‖≃I/|j–k|p, j ≠ k, 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 〈A〉φ(t) ≤ ctδ, where δ>0 depends on p and μ. Typical applications concern the energy expectation 〈H0〉φ(t in case H0(t) ≡ H0 or 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 H0(t) is a time-dependent multiplicative potential.

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