[Paper Review] Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

Sylvain Golénia|arXiv (Cornell University)|Jun 3, 2011

Spectral Theory in Mathematical Physics21 references6 citations

TL;DR

This paper establishes a discrete analog of the Hardy inequality for magnetic discrete Laplacians on locally finite graphs, characterizes the form-domain, and derives the asymptotic eigenvalue distribution under perturbations. It proves that the essential spectrum of the perturbed Laplacian coincides with that of the reference tree Laplacian and shows that eigenvalues interlace asymptotically with those of the weighted degree operator, with a convergence rate of 1 under suitable decay conditions on the perturbation.

ABSTRACT

In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.

Motivation & Objective

• To establish a discrete version of the Hardy inequality for magnetic discrete Laplacians on weighted, locally finite graphs.
• To characterize the form-domain of the magnetic discrete Laplacian and identify conditions under which the essential spectrum is absent.
• To derive the asymptotic distribution of eigenvalues for perturbed discrete Laplacians, particularly in relation to the weighted degree operator.
• To prove the stability of the essential spectrum under potential perturbations that decay relative to the degree growth.

Proposed method

• The authors define a quadratic form associated with the magnetic discrete Laplacian and use the Friedrichs extension to construct a self-adjoint operator.
• They introduce a modified isoperimetric constant α(G) to relate the Laplacian to the weighted degree operator, improving the standard form inequality.
• A key technical tool is a discrete Hardy-type inequality that minorates the Laplacian by a weighted degree operator minus a potential term, independent of the magnetic field.
• The proof of asymptotic eigenvalue equivalence relies on the min-max principle and resolvent comparison via the KLMN theorem and Helffer-Sj¨ostrand formula.
• The stability of the essential spectrum is shown by proving the relative compactness of the resolvent difference using decay conditions on the perturbation.
• The analysis is extended to bi-partite graphs and trees, where stronger domain and spectral control is possible.

Experimental results

Research questions

• RQ1 Under what conditions does the magnetic discrete Laplacian on a graph have a well-defined form-domain, and when is the essential spectrum empty?
• RQ2 How does the asymptotic distribution of eigenvalues of a perturbed discrete Laplacian relate to the weighted degree operator?
• RQ3 Can the essential spectrum of a perturbed discrete Laplacian be characterized in terms of the reference graph’s Laplacian?
• RQ4 What role does the isoperimetric constant α(G) play in controlling the spectral behavior of the Laplacian?
• RQ5 How do perturbations that grow sublinearly relative to the degree affect the spectral properties of the discrete Laplacian?

Key findings

• The form-domain of the magnetic discrete Laplacian coincides with that of the weighted degree operator if and only if the isoperimetric constant α(G) > 0.
• The essential spectrum of the Friedrichs extension of the perturbed Laplacian HF is equal to that of the reference tree Laplacian ∆E◦,θ◦.
• If the weighted degree dG◦(x) → ∞ as |x| → ∞, then the essential spectrum of HF is empty.
• The N-th eigenvalue of HF and the N-th eigenvalue of dG◦(Q) satisfy limN→∞ λN(HF)/λN(dG◦(Q)) = 1, indicating asymptotic interlacing.
• The perturbation V and edge weight difference Λ must satisfy |V(x)| + Λ(x) = o(1 + dG◦(x)) as |x| → ∞ for spectral stability.
• The discrete Hardy inequality provides a lower bound ⟨f, Vm(Q)f⟩ ≤ ⟨f, ∆E,θf⟩ with Vm(x) = dG(x) − Wm(x), independent of the magnetic field.