Spectral comparison results for Laplacians on discrete graphs

TL;DR

The article develops a spectral comparison framework for Laplacians on discrete graphs, including infinite graphs, by comparing a baseline realization $L_0$ with a perturbed realization $L_c$ defined via a potential $c$ on $(X,m)$. The authors prove a discrete local Weyl law and, crucially, a main spectral-perturbation identity: $\sum_{n=1}^{\dim \ell^2(X,m)} (\lambda_n(c)-\lambda_n(0)) = \sum_{x \in X} \frac{c(x)}{m(x)}$ under $\frac{c}{m}\in \ell^1(X,\mathbf{1})$, establishing that eigenvalue shifts are governed by the perturbation. A Hadamard-type derivative formula for $\lambda_n(\tau c)$ with respect to the perturbation parameter $\tau$ is derived, and the approach uses a discrete heat kernel $p_t^c$ together with Mercer's theorem and dominated convergence. These results imply asymptotic isospectrality on infinite graphs when the perturbation is summable and lead to an Ambarzumian-type theorem, connecting spectral data to the perturbation and inverse spectral theory in the discrete setting.

Abstract

In the recent literature, various authors have studied spectral comparison results for Schrödinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note we derive such spectral comparison results in a rather general framework for general and possibly infinite discrete graphs. Along the way, we establish a discrete version of the local Weyl law whose proof does neither involve any Tauberian theorem nor the Weyl law as used in the continuous case.

Figures (1)

• Figure 1: The infinite path graph $b$ over $(\mathbb{N},m)$.

Theorems & Definitions (14)

• Remark 1
• Lemma 3
• proof
• Example 4
• Theorem 5: Spectral comparison for discrete graphs
• Remark 6
• Example 7
• Proposition 8: Local Weyl law
• proof
• Lemma 9