Computing the spectrum and pseudospectrum of infinite-volume operators from local patches

TL;DR

The paper develops a rigorous framework to compute the spectrum and the \(\\varepsilon\)-pseudospectrum of short-range infinite-volume operators with finite local complexity, using only local patches and explicit error bounds. By reducing to finite-range operators and exploiting the lower norm \(\\rho_H(\\lambda)\\), it provides computable upper and lower bounds, enabling both spectrum and pseudospectrum computations with guaranteed Hausdorff accuracy. A central result is a spectral gap bound that links patch-based approximations to true gaps, together with a computability theory placing the flc problem in the SCI-Delta_1 class, in contrast to general negative results. The framework also yields practical algorithms for non-normal operators, including a computable \(\\varepsilon\)-pseudospectrum via uneven sections and adaptive grid refinement, with termination guarantees. These contributions advance reliable spectral analysis for physically relevant models such as discrete Schrödinger operators and Hofstadter-type systems on infinite graphs.

Abstract

We show how the spectrum of normal discrete short-range infinite-volume operators can be approximated with two-sided error control using only data from finite-sized local patches. As a corollary, we prove the computability of the spectrum of such infinite-volume operators with the additional property of finite local complexity and provide an explicit algorithm. Such operators appear in many applications, e.g. as discretizations of differential operators on unbounded domains or as so-called tight-binding Hamiltonians in solid state physics. For a large class of such operators, our result allows for the first time to establish computationally also the absence of spectrum, i.e. the existence and the size of spectral gaps. We extend our results to the $\varepsilon$-pseudospectrum of non-normal operators, proving that also the pseudospectrum of such operators is computable.

Figures (1)

• Figure 1: Exact computation of the $\varepsilon$-pseudospectrum of a non-Hermitian Hamiltonian with a cut-and-project potential for $\varepsilon = 0.5$. The Hamiltonian is defined by $H\psi(n) = -\psi(n-1) + V(n)\psi(n) - \psi(n+1)$. The chosen potential has the form $V(n) = (1 + i)\mathbf{1}(\alpha n < 1 / \alpha)$, where we chose $\alpha = 1.66$ (For $\alpha = (1 + \sqrt 5) / 2$, this construction gives the Fibonacci quasicrystal, but we chose $\alpha = 1.66$ because it creates a less uniform potential leading to a slower convergence and thus a more pronounced effect of increasing $L$ in the pictures.). The left and right column show the same computation with $L = 20$ and $L = 300$, respectively. The uppermost row shows the lower, spectral gap bound on $\rho_H$, while the row below shows the upper bound on $\rho_H$ from colbrook_2019. If the lower bound is positive, the associated point is known to be in the complement of the pseudospectrum; if the upper bound is negative, the point is known to be inside the pseudospectrum. This gives a decomposition of the plane into three sets $R, U,$ and $S$ of points, where it is known that $S \subseteq \mathop{\mathrm{Spec}}\nolimits_\varepsilon(H)$, that $R \cap \mathop{\mathrm{Spec}}\nolimits_\varepsilon(H) = \varnothing$, and no statement can be made about $U$. This is very similar to the sets $S_\tau$, $R_\tau$ and $U_\tau$ in Section \ref{['sec-computability-pseudospectrum']}, except that here we have fixed $L$ instead of $\tau$ and we do not vary the spacing of the grid on which $\tilde{\rho}_H(\lambda, \tau)$ is evaluated.

Theorems & Definitions (49)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Definition 2.9