Time dependent quantum systems have become indispensable in science and itsapplications, particularly at the atomic and molecular levels. Here, we discussthe approximation of closed time dependent quantum systems on bounded domains,via iterative methods in Sobolev space based upon evolution operators.Recently, existence and uniqueness of weak solutions were demonstrated by acontractive fixed point mapping defined by the evolution operators. Convergentsuccessive approximation is then guaranteed. This article uses the same mappingto define quadratically convergent Newton and approximate Newton methods.Estimates for the constants used in the convergence estimates are provided. Theevolution operators are ideally suited to serve as the framework for thisoperator approximation theory, since the Hamiltonian is time dependent. Inaddition, the hypotheses required to guarantee quadratic convergence of theNewton iteration build naturally upon the hypotheses used for theexistence/uniqueness theory.
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