The Spectral Edges Conjecture via Corners

2025/10/11 by Faust, Matthew, Sottile, Frank · 1 citation
#05C50 #14Q20 #47A75 #81Q10 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2510.10143

Abstract

The Spectral Edges Conjecture is a well-known and widely believed conjecture in the theory of discrete periodic operators. It states that the extrema of the dispersion relation are isolated, non-degenerate, and occur in a single band. We present two infinite families of periodic graphs which satisfy the Spectral Edges Conjecture. For each, every extremum of the dispersion relation is a corner point (point of symmetry). In fact, each spectral band function is a perfect Morse function. We also give a construction that increases dimension, while preserving that each spectral band function is a perfect Morse function.

Citations

• Absence of flat bands for discrete periodic graph operators with generic potentials
• Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs
• Algebraic Aspects of Periodic Graph Operators
• Critical points of discrete periodic operators
• An overview of periodic elliptic operators
• Schrödinger operators on periodic discrete graphs
• The Creation of Spectral Gaps by Graph Decoration

Cited by

• The Critical Point Degree of a Periodic Graph
• Oscillation of graph eigenfunctions

Related

• Critical points of discrete periodic operators
• Asymptotic isospectrality of Schrödinger operators on periodic graphs
• Schrödinger operators on periodic discrete graphs
• Spectral band bracketing for Laplacians on periodic metric graphs
• Three conjectures in extremal spectral graph theory