The Spectral Edges Conjecture via Corners
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915546529792000 |
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| author | Faust, Matthew Sottile, Frank |
| author_facet | Faust, Matthew Sottile, Frank |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Spectral Edges Conjecture via Corners Faust, Matthew Sottile, Frank Spectral Theory Algebraic Geometry Combinatorics 47A75, 81Q10, 05C50, 14Q20 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. |
| title | The Spectral Edges Conjecture via Corners |
| topic | Spectral Theory Algebraic Geometry Combinatorics 47A75, 81Q10, 05C50, 14Q20 |
| url | https://arxiv.org/abs/2510.10143 |