A Generalization of the Arzelà–Ascoli Theorem and its Application to Dyson Perturbation Theory

An approximate sequence Un(t, t0) for the time-development operator U(t,t0) has been constructed, when n→∞ it approaches the Dyson series. A generalization of the Arzelà–Ascoli theorem for a set of continuous functions to a set of operators is given. We show that if the norm of H(t) in Equation (1.1) is a Lebesgue integrable function with respect to time t, then the Dyson series is uniformly weak convegent and Equation (1.1) has a unique solution. The preliminary application of the above results to quantum mechanics is studied.