The Quantum Faedo-Galerkin Equation: Evolution Operator and Time Discretization

Time dependent quantum systems have become indispensable in science and nanotechnology. Disciplines including chemical physics and electrical engineering have used approximate evolution operators to solve these systems for targeted physical quantities. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains via evolution operators. The work builds upon the use of weak solutions, which includes a framework for the evolution operator based upon dual spaces. We are able to derive the corresponding Faedo-Galerkin equation as well as its time discretization, yielding a fully discrete theory. We obtain corresponding approximation estimates. These estimates make no regularity assumptions on the weak solutions, other than their inherent properties. Of necessity, the estimates are in the dual norm, which is natural for weak solutions. This appears to be a novel aspect of this approach.