### 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 language | English (US) |
---|---|

Journal | Journal of Numerical Mathematics |

DOIs | |

State | Accepted/In press - Feb 19 2018 |

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### 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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**A Tight Nonlinear Approximation Theory for Time Dependent Closed Quantum Systems.** / Jerome, Joseph W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Tight Nonlinear Approximation Theory for Time Dependent Closed Quantum Systems

AU - Jerome, Joseph W.

PY - 2018/2/19

Y1 - 2018/2/19

N2 - 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.

AB - 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.

KW - Faedo-Galerkin approximations

KW - Time dependent quantum systems

KW - numerical fixed points

KW - time-ordered evolution operators

UR - http://www.scopus.com/inward/record.url?scp=85042686552&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042686552&partnerID=8YFLogxK

U2 - 10.1515/jnma-2017-0128

DO - 10.1515/jnma-2017-0128

M3 - Article

AN - SCOPUS:85042686552

JO - Journal of Numerical Mathematics

JF - Journal of Numerical Mathematics

SN - 1570-2820

ER -