Core Concepts
The paper proposes a new algorithm, the irrational-window-filter projection method (IWFPM), for efficiently solving arbitrary dimensional global quasiperiodic systems, especially quasiperiodic Schrödinger eigenproblems. IWFPM utilizes the concentrated distribution of Fourier coefficients to filter out relevant spectral points using an irrational window and employs an index-shift transform to enable the use of Fast Fourier Transform.
Abstract
The paper introduces the concept of quasiperiodic functions and the projection method (PM), which is a widely used algorithm for solving quasiperiodic systems. It then presents the key ideas behind the proposed irrational-window-filter projection method (IWFPM):
Segment 1:
- IWFPM is based on the projection method, but further utilizes the concentrated distribution of Fourier coefficients in quasiperiodic systems.
- It employs an irrational window to filter out relevant spectral points and a corresponding index-shift transform to make the Fast Fourier Transform available.
- The error analysis on the function approximation level is provided, showing that IWFPM can achieve consistent convergence by adjusting the size of the irrational window.
Segment 2:
- The paper applies IWFPM to solve 1D, 2D, and 3D quasiperiodic Schrödinger eigenproblems (QSEs) and demonstrates its accuracy and efficiency.
- For both extended and localized quantum states, IWFPM exhibits a significant computational advantage over the projection method.
- An efficient diagonal preconditioner is designed for the discrete QSEs to significantly reduce the condition number.
Segment 3:
- The widespread existence of the concentrated Fourier coefficient distribution feature can endow IWFPM with significant potential for broader applications in quasiperiodic systems.
- The paper concludes by summarizing the key contributions and outlining future research directions.
Stats
The paper does not contain any explicit numerical data or statistics to support the key logics. The analysis is based on the theoretical development of the IWFPM algorithm and its application to quasiperiodic Schrödinger eigenproblems.
Quotes
The paper does not contain any striking quotes that support the key logics.