### Abstract

Time dependent quantum systems have become indispensable in science and its applications, particularly at the atomic and molecular levels. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains, via iterative methods in Sobolev space based upon evolution operators. Recently, existence and uniqueness of weak solutions were demonstrated by a contractive fixed point mapping defined by the evolution operators. Convergent successive approximation is then guaranteed. This article uses the same mapping to define quadratically convergent Newton and approximate Newton methods. Estimates for the constants used in the convergence estimates are provided. The evolution operators are ideally suited to serve as the framework for this operator approximation theory, since the Hamiltonian is time-dependent. In addition, the hypotheses required to guarantee quadratic convergence of the Newton iteration build naturally upon the hypotheses used for the existence/uniqueness theory.

Original language | English (US) |
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Pages (from-to) | 130-147 |

Number of pages | 18 |

Journal | Nonlinear Analysis: Real World Applications |

Volume | 40 |

DOIs | |

Publication status | Published - Apr 2018 |

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### Keywords

- Newton iteration
- Quadratic convergence
- Time dependent quantum systems
- Time-ordered evolution operators

### ASJC Scopus subject areas

- Analysis
- Engineering(all)
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics