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准周期量子动力系统的数值求解

李雪阳, 蒋凯   

  1. 湘潭大学数学与计算科学学院, 科学工程计算与数值仿真湖南省重点实验室, 湘潭 411105
  • 收稿日期:2020-08-31 发布日期:2021-03-18
  • 通讯作者: 蒋凯,kaijiang@xtu.edu.cn.
  • 基金资助:
    国家自然科学基金(11771368),湖南省教育厅重点项目(19A500),湖南省教育厅优秀青年项目(20B566)资助.

李雪阳, 蒋凯. 准周期量子动力系统的数值求解[J]. 数值计算与计算机应用, 2021, 42(1): 3-17.

Li Xueyang, Jiang Kai. NUMERICAL SIMULATION FOR QUASIPERIODIC QUANTUM DYNAMICAL SYSTEMS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(1): 3-17.

NUMERICAL SIMULATION FOR QUASIPERIODIC QUANTUM DYNAMICAL SYSTEMS

Li Xueyang, Jiang Kai   

  1. Department of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Enginerring, Xiangtan University, Xiangtan 411105, China
  • Received:2020-08-31 Published:2021-03-18
本文提出了研究准周期量子动力系统的计算方法.传统计算方法通常使用大的周期系统来近似计算准周期系统,这会产生丢番图逼近误差.我们的方法将准周期系统在高维周期结构中表示和计算,这样不仅可以避免丢番图逼近误差,而且可以将周期系统的高效算法,比如快速Fourier变换直接运用到新的算法中.我们将算法用于计算具有准周期势的线性薛定谔方程和准周期初值的非线性薛定谔方程.数值结果表明了算法的可靠性和高效性,并可以用于研究包含Anderson局域化、非线性光子准晶在内的准周期量子行为.
We propose a novel numerical method for computing the quasiperiodic quantum dynamical systems. Conventional approach common uses a large periodic cell to approximate the quasiperiodic system which produces the Diophantine approximation error. Our algorithm represents the quasiperiodic system in higher dimensional periodic structure, which can avoid the Diophantine approximation error and can use the fast Fourier transformation method for highly efficient computation. We apply the approach to the compute the linear Schrödinger equation with quasiperiodic potential and nonlinear Schrödinger equation with quasiperiodic initial function. Numerical results show the reliability and efficiency of our algorithm, and the ability of studying the rich quasiperodic quantum behaviors, including Anderson localization and nonlinear photonic quasicrystals.

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