Efficient Numerical Method for Solving Arbitrary Dimensional Quasiperiodic Schrödinger Eigenproblems

Concepts de base

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.

Résumé

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.

Citations

The paper does not contain any striking quotes that support the key logics.

Idées clés tirées de

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

by Kai Jiang,Xu... à arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04507.pdf

Questions plus approfondies

What other types of quasiperiodic systems, beyond Schrödinger eigenproblems, could benefit from the IWFPM approach

The IWFPM approach, designed for solving quasiperiodic Schrödinger eigenproblems, can be beneficial for various other types of quasiperiodic systems. One such application could be in the field of quasicrystals, where the ordered but non-repeating atomic structures exhibit quasiperiodic characteristics. By applying the IWFPM method to quasicrystal systems, researchers can efficiently analyze the electronic properties and energy levels of these materials. Additionally, the method could be extended to study quasiperiodic magnetic structures, such as magnetic quasicrystals, to understand their unique magnetic properties and behavior.

How can the IWFPM method be further extended or generalized to handle even more complex quasiperiodic structures or higher-dimensional problems

To handle more complex quasiperiodic structures or higher-dimensional problems, the IWFPM method can be extended in several ways. One approach could involve incorporating adaptive algorithms that dynamically adjust the size and shape of the concentrated Fourier coefficient area based on the system's properties. This adaptability would enhance the method's efficiency and accuracy when dealing with intricate quasiperiodic systems. Furthermore, the IWFPM approach could be generalized to include non-linear quasiperiodic systems by developing specialized filtering techniques to handle the non-linearities effectively. Additionally, for higher-dimensional problems, the method could be optimized to leverage parallel computing capabilities to expedite the computations and address the increased complexity of multi-dimensional systems.

Are there any potential connections between the concentrated Fourier coefficient distribution observed in quasiperiodic systems and the underlying mathematical properties or physical phenomena of these systems

The concentrated distribution of Fourier coefficients observed in quasiperiodic systems can be indicative of underlying mathematical properties and physical phenomena. In the context of Schrödinger eigenproblems, the concentration of Fourier coefficients within specific areas reflects the system's quasiperiodic nature and the localized or extended states of the wavefunctions. This distribution feature can be linked to the system's spectral properties, such as the presence of pure point, singular continuous, or absolutely continuous spectra. Moreover, the concentrated Fourier coefficients may signify the system's sensitivity to specific frequencies or spatial arrangements, providing insights into the system's stability, localization behavior, and spectral characteristics. By studying the relationship between the concentrated Fourier coefficients and the system's properties, researchers can gain a deeper understanding of the quasiperiodic structures and their implications in various physical phenomena.