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Irrational-window-filter projection method and application to quasiperiodic Schrödinger eigenproblems

Kai Jiang, Xueyang Li, Yao Ma, Juan Zhang, Pingwen Zhang, Qi Zhou

Abstract

In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to 1D, 2D, and 3D quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically verified.

Irrational-window-filter projection method and application to quasiperiodic Schrödinger eigenproblems

Abstract

In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to 1D, 2D, and 3D quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically verified.
Paper Structure (14 sections, 5 theorems, 78 equations, 8 figures, 9 tables, 2 algorithms)

This paper contains 14 sections, 5 theorems, 78 equations, 8 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Quasiperiodic functions and projection method (PM)
  3. Irrational-window-filter projection method (IWFPM)
  4. Irrational window
  5. Index-shift map
  6. Implementation
  7. Error analysis
  8. Application to quasiperiodic Schrödinger eigenproblems (QSEs)
  9. IWFPM discretization
  10. Numerical experiments
  11. Conclusion and outlook
  12. Proof of \ref{['lm:norm relation']}
  13. Proof of \ref{['th:proj']}
  14. Proof of \ref{['th:int']}

Key Result

Lemma 2.3

For a $d$-dimensional quasiperiodic function $u$ and its associated parent function $U$, it holds $\hat{u}_{\bm{q}}=\hat{{U}}_{\bm{k}}$ when $\bm{q}=\bm{Pk}$.

Figures (8)

  • Figure 1: Rectangle index set $\mathcal{K}_N$, parallelogram index set $\mathcal{K}_{K,L}$, and irrational window $\mathcal{W}_{K,L}$ when $d=1$, $n=2$, $N=12$, $K=2$, $L=6$, $\bm{P}=(1,(\sqrt{5}+1)/2)$.
  • Figure 1: Results of solving 1D QSE with potential \ref{['eq:E1_equation']} by IWFPM. The top row: probability density function $\bm{\rho}$; The bottom row: Fourier coefficients $\tilde{U}_{\bm{k}}$.
  • Figure 2: The parallelogam index set $\mathcal{K}_{K,L}$ (left), the rectangle index set $\mathcal{K}^*_{K,L}$ (middle), and the grid points $\mathcal{G}_{K,L}$ (right) when $d=1$, $n=2$, $K=2$, $L=6$, $\bm{P}=(1,(\sqrt{5}+1)/2)$.
  • Figure 2: Required DOFs of PM and IWFPM when they achieve the same accurate $E_v$ for solving 1D QSE with potential \ref{['eq:E1_equation']} ($v_0=2.5$).
  • Figure 3: Results of solving 2D QSE with potential \ref{['eq:E2_equation']} by IWFPM. The top row: probability density function $\bm{\rho}$; The bettom row: Fourier coefficients $\tilde{U}_{\bm{k}}~(\tilde{U}_{\bm{k}}\geq10^{-8})$.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Lemma 2.3: jiang2024numerical, Theorem 4.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.1
  • Proof 1
  • ...and 12 more