Title: Convergent Iteration in Sobolev Space for Time Dependent Closed Quantum Systems,
Abstract: Time dependent quantum systems have become indispensable in science and its
applications, particularly at the atomic and molecular levels. Here, we discuss
the 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 a
contractive fixed point mapping defined by the evolution operators. Convergent
successive approximation is then guaranteed. This article uses the same mapping
to define quadratically convergent Newton and approximate Newton methods.
Estimates for the constants used in the convergence estimates are provided. The
evolution operators are ideally suited to serve as the framework for this
operator approximation theory, since the Hamiltonian is time dependent. In
addition, the hypotheses required to guarantee quadratic convergence of the
Newton iteration build naturally upon the hypotheses used for the
existence/uniqueness theory.