Core Concepts

The authors present a general algorithm to compute the spectrum and pseudospectrum of short-range, discrete, infinite-volume operators with finite local complexity, with rigorous error control.

Abstract

The paper addresses the problem of efficiently computing the spectrum and pseudospectrum of infinite-volume operators, which is a fundamental problem in mathematical physics with applications in areas like solid state physics and the study of differential operators on unbounded domains.
The key insights are:
For operators with finite local complexity (flc), the infimum of the smallest singular value of certain finite-dimensional submatrices provides a quantitative lower bound on the lower norm function ρH, which is crucial for establishing computability.
By combining this lower bound with the known upper bound from the "uneven sections" method, the authors show that the spectrum and pseudospectrum of flc operators are computable with error control in Hausdorff distance.
The authors prove that the flc spectral problem is in the computability class ∆1, meaning it can be solved with full error control, in contrast to the general spectral problem which is only in the weaker class ∆2 or Σ1.
The algorithm can be applied to a wide range of infinite-volume operators arising in physics, including discrete Schrödinger operators with potentials defined by substitution rules, the Hofstadter model, and random operators, for which the spectrum and pseudospectrum were not previously known to be computable.

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Key Insights Distilled From

by Paul Hege,Ma... at **arxiv.org** 03-29-2024

Deeper Inquiries

To improve the practical efficiency of the algorithm, especially for higher-dimensional systems, several strategies can be implemented. One approach could involve optimizing the enumeration of local patches by utilizing advanced data structures and algorithms tailored for efficient search and retrieval. Implementing parallel processing techniques could also enhance the algorithm's speed by distributing the computational workload across multiple processors or cores. Additionally, leveraging machine learning algorithms for pattern recognition within the operators could help streamline the identification of equivalent action subsets, reducing the computational complexity of the problem. Furthermore, exploring domain-specific optimizations based on the specific characteristics of the operators being analyzed could lead to significant efficiency gains.

Beyond finite local complexity, other structural properties of operators could be leveraged to expand the class of computable infinite-volume spectral problems. For example, exploiting symmetries present in the operators could lead to more efficient algorithms by reducing the search space for equivalent action subsets. Utilizing sparsity patterns in the operator matrices could also enable more targeted computations, focusing on the most relevant areas of the operator that contribute significantly to the spectrum. Additionally, incorporating domain-specific knowledge or constraints into the algorithm could further refine the spectral computations, tailoring the approach to the specific characteristics of the operators under study.

The results presented by the authors have significant implications for the study of the spectral properties of aperiodic systems and quasicrystals. By demonstrating the computability of the spectrum and pseudospectrum for operators with finite local complexity, the authors have opened up new avenues for analyzing the spectral behavior of complex systems with aperiodic structures. This computability allows for a more systematic and rigorous investigation of spectral gaps, eigenvalues, and pseudospectral properties in a wide range of aperiodic systems, including quasicrystals. The ability to computationally analyze the spectral properties of such systems provides valuable insights into their behavior, paving the way for further advancements in the understanding of aperiodic materials and structures.

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