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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.09731 |
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| _version_ | 1866929214153818112 |
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| author | Faust, Matthew Liu, Wencai Matos, Rodrigo Plute, Jenna Robinson, Jonah Tao, Yichen Tran, Ethan Zhuang, Cindy |
| author_facet | Faust, Matthew Liu, Wencai Matos, Rodrigo Plute, Jenna Robinson, Jonah Tao, Yichen Tran, Ethan Zhuang, Cindy |
| contents | Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in (\mathbb{Z}^+)^d$ for each $j\in \{1,\ldots,d\}$, and denote by $Δ$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. Using Macaulay2, we first numerically find complex-valued $Γ$-periodic potentials $V:\mathbb{Z}^d\to \mathbb{C}$ such that the operators $Δ+V$ and $Δ$ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09731 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators Faust, Matthew Liu, Wencai Matos, Rodrigo Plute, Jenna Robinson, Jonah Tao, Yichen Tran, Ethan Zhuang, Cindy Spectral Theory Mathematical Physics Combinatorics Primary 58J53. Secondary: 47B36, 35P05, 35J10 Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in (\mathbb{Z}^+)^d$ for each $j\in \{1,\ldots,d\}$, and denote by $Δ$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. Using Macaulay2, we first numerically find complex-valued $Γ$-periodic potentials $V:\mathbb{Z}^d\to \mathbb{C}$ such that the operators $Δ+V$ and $Δ$ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions. |
| title | Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators |
| topic | Spectral Theory Mathematical Physics Combinatorics Primary 58J53. Secondary: 47B36, 35P05, 35J10 |
| url | https://arxiv.org/abs/2401.09731 |