A Tight Nonlinear Approximation Theory for Time Dependent Closed Quantum Systems

Joseph W. Jerome*

*Corresponding author for this work

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
JournalJournal of Numerical Mathematics
DOIs
StateAccepted/In press - Feb 19 2018

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Approximation theory
Hamiltonians
Nonlinear Approximation
Approximation Theory
Galerkin
Quantum Systems
Fixed point
Drift-diffusion Model
Closed
Sobolev spaces
Evolution Operator
Projection Operator
Sobolev Spaces
Mathematical operators
Differential operator
Semiconductors
Bounded Domain
Calculus
Kinetics
Subspace

Keywords

  • Faedo-Galerkin approximations
  • Time dependent quantum systems
  • numerical fixed points
  • time-ordered evolution operators

ASJC Scopus subject areas

  • Computational Mathematics

Cite this

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