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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by Paul Hege,Ma... pada arxiv.org 03-29-2024
https://arxiv.org/pdf/2403.19055.pdfPertanyaan yang Lebih Dalam