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arXiv · 2510.10143

The Spectral Edges Conjecture via Corners

Matthew Faust·Frank Sottile
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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.

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Matthew Faust, Frank Sottile. 2025-10-11. The Spectral Edges Conjecture via Corners. https://arxiv.org/abs/2510.10143

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