The Spectral Edges Conjecture via Corners

TL;DR

The paper investigates the spectral edges conjecture for discrete periodic operators by constructing two infinite families of ${\mathbb Z}^d$-periodic graphs in which every spectral band extremum is a corner point and each band function is a perfect Morse function. Using Floquet theory and the Bloch variety, it shows that for minimally sparse graphs and isthmus-connected graphs (including periodic flower graphs), all critical points are nondegenerate corner points under generic labeling, implying the conjecture holds. A coordinate-projection technique and algebraic arguments (Sylvester resultants) are employed to rule out noncorner critical points and flat bands in generic cases. The dimension-raising parallel extension construction demonstrates that the perfect Morse property is preserved under increasing ambient dimension. Collectively, these results map structural graph conditions to robust spectral-edge nondegeneracy in higher dimensions, enriching discrete spectral theory with explicit graph families satisfying the conjecture.

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.

Figures (4)

• Figure 1: Three dispersion relations for graphs with $d=2$ and $|W|=2,3,4$.
• Figure 2: The Lieb lattice and two of its dispersion relations.
• Figure 3: A ${\mathbb Z}^2$-periodic minimally sparse graph.
• Figure 4: Three Bloch varieties with their parameters.

Theorems & Definitions (21)

• Theorem 1.1
• proof
• Lemma 1.2
• proof
• Proposition 2.1
• Theorem 2.2
• proof : Proof of Theorem \ref{['Thm:RealFamCross']}
• Corollary 2.3
• proof
• proof