High-Accuracy Numerical Methods and Convergence Analysis for Schrödinger Equation with Incommensurate Potentials

Numerical solving the Schrödinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose two high-accuracy numerical methods to solve the time-dependent...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 101; no. 1; p. 18
Main Authors: Jiang, Kai, Li, Shifeng, Zhang, Juan
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2024
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Convergence
Eigenvalues
Fourier transforms
Localization
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Numerical analysis
Numerical methods
Ordinary differential equations
Schrodinger equation
Spectral methods
Theoretical
Time dependence
Two dimensional analysis
Projection method
Quasiperiodic spectral method
68W40
65T40
42A75
Quasiperiodic Schrödinger equation
74S25
Second-order operator splitting method
Convergence analysis
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:Numerical solving the Schrödinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose two high-accuracy numerical methods to solve the time-dependent quasiperiodic Schrödinger equation. Concretely, we discretize the spatial variables by the quasiperiodic spectral method and the projection method, and the time variable by the second-order operator splitting method. The corresponding convergence analysis is also presented and shows that the proposed methods both have spectral convergence rates in space and second order accuracy in time, respectively. Meanwhile, we analyse the computational complexity of these numerical algorithms. One- and two-dimensional numerical results verify these convergence conclusions, and demonstrate that the projection method is more efficient.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02658-3