A Tight Nonlinear Approximation Theory for Time Dependent Closed Quantum Systems

TL;DR

This paper extends nonlinear approximation theory to time-dependent quantum systems with external potentials, demonstrating convergence of Galerkin solutions in Sobolev space at optimal rates.

Contribution

It applies a classical fixed point approximation theory to quantum systems with evolving Hamiltonians, including nonlinear and external potentials, via evolution operators.

Findings

Existence of unique Faedo-Galerkin solutions for quantum systems.

Convergence of solutions in Sobolev space at maximal projection rates.

Application of the theory to bounded domain quantum models.

Abstract

The approximation of fixed points by numerical fixed points was presented in the elegant monograph of Krasnosel'skii et al. (1972). The theory, both in its formulation and implementation, requires a differential operator calculus, so that its actual application has been selective. The writer and Kerkhoven demonstrated this for the semiconductor drift-diffusion model in 1991. In this article, we show that the theory can be applied to time dependent quantum systems on bounded domains, via evolution operators. In addition to the kinetic operator term, the Hamiltonian includes both an external time dependent potential and the classical nonlinear Hartree potential. Our result can be paraphrased as follows: For a sequence of Galerkin subspaces, and the Hamiltonian just described, a uniquely defined sequence of Faedo-Galerkin solutions exists; it converges in Sobolev space, uniformly in time, at the maximal rate given by the projection operators.