Measure of the spectra of periodic graph operators in the large-coupling limit

TL;DR

The paper addresses the problem of when the spectrum of a periodic discrete Schrödinger operator $H_{\mu Q}$ on a ${\mathbb Z}^d$-periodic graph collapses in the large-coupling limit. It combines Floquet theory, a quotient-graph cohomology framework, and perturbation theory to derive a sharp topological criterion: ${\mathrm Leb}({\mathrm spec}(H_{\mu Q})) \to 0$ as $\mu \to \infty$ if and only if there is no infinite connected path along which the potential $Q$ is constant, i.e., no infinite $Q$-flat path. This criterion has an equivalent formulation in terms of the graph cohomology of the quotient graph and a local energy description near energies $\mu a$ with $a$ in the range of $Q$, and it extends prior results to degenerate potentials. The findings provide a precise topological condition for spectral collapse, with implications for transport properties in periodic media and potential explicit lower bounds in special lattice cases.

Abstract

We derive a sharp criterion on the spectra of periodic discrete Schrödinger operators acting on connected periodic lattices: the measure of the spectrum goes to zero as the coupling constant goes to infinity if and only if there is no infinite connected path of degeneracies.

Theorems & Definitions (6)

• Theorem 1.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['t:perMeasGen']}