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

TL;DR

This paper establishes a precise criterion for when the spectrum measure of periodic discrete Schrödinger operators diminishes to zero in the large-coupling limit, linking it to the absence of infinite degeneracy paths.

Contribution

It provides a sharp, necessary and sufficient condition connecting spectral measure decay to the structure of degeneracies in periodic lattice operators.

Findings

Spectrum measure tends to zero at large coupling if no infinite degeneracy path exists.

The criterion is both necessary and sufficient for spectral measure vanishing.

Connects spectral properties with geometric degeneracy structures in periodic lattices.

Abstract

We derive a sharp criterion on the spectra of periodic discrete Schr\"odinger 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.