### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 590-601 |

Number of pages | 12 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 38 |

Issue number | 5 |

DOIs | |

State | Published - May 4 2017 |

### Fingerprint

### Keywords

- Approximation estimates
- Faedo-Galerkin operator equation
- time dependent quantum systems
- time-ordered evolution operators

### ASJC Scopus subject areas

- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization

### Cite this

}

**The Quantum Faedo-Galerkin Equation : Evolution Operator and Time Discretization.** / Jerome, Joseph W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Quantum Faedo-Galerkin Equation

T2 - Evolution Operator and Time Discretization

AU - Jerome, Joseph W

PY - 2017/5/4

Y1 - 2017/5/4

N2 - 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.

AB - 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.

KW - Approximation estimates

KW - Faedo-Galerkin operator equation

KW - time dependent quantum systems

KW - time-ordered evolution operators

UR - http://www.scopus.com/inward/record.url?scp=85015776895&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015776895&partnerID=8YFLogxK

U2 - 10.1080/01630563.2016.1252393

DO - 10.1080/01630563.2016.1252393

M3 - Article

AN - SCOPUS:85015776895

VL - 38

SP - 590

EP - 601

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 5

ER -