Convergent Iteration in Sobolev Space for Time Dependent Closed Quantum
By: Joseph W. Jerome
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.
This paper has been published online, Jour. Nonlinear Anal.: Real World
Applications, 40 (2018), 130-147.
It can also be accessed at: arXiv: 1706.09788.