Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908822384148480 |
|---|---|
| author | Faust, Matthew Friedman, Leo O'Malley, Gavin Ramos, Rolando Sharma, Aaryan |
| author_facet | Faust, Matthew Friedman, Leo O'Malley, Gavin Ramos, Rolando Sharma, Aaryan |
| contents | Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in \mathbb{Z}^+$ for each $j\in \{1,\ldots,d\}$, and denote by $Δ$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. We describe various algebraic properties of the ideal of spectral invariants for the discrete Laplacian when $d=1$, including a construction of a Gröbner basis. We also present various collections of complex $Γ$-periodic potentials $V$ that are such that $Δ$ and $Δ+ V$ are Floquet isospectral. We end with a discussion of the general setting, where the $q_i$ are taken to be vectors in $\mathbb{Z}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08304 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian Faust, Matthew Friedman, Leo O'Malley, Gavin Ramos, Rolando Sharma, Aaryan Spectral Theory Mathematical Physics Algebraic Geometry 15A29, 47B36, 81Q35 Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in \mathbb{Z}^+$ for each $j\in \{1,\ldots,d\}$, and denote by $Δ$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. We describe various algebraic properties of the ideal of spectral invariants for the discrete Laplacian when $d=1$, including a construction of a Gröbner basis. We also present various collections of complex $Γ$-periodic potentials $V$ that are such that $Δ$ and $Δ+ V$ are Floquet isospectral. We end with a discussion of the general setting, where the $q_i$ are taken to be vectors in $\mathbb{Z}^d$. |
| title | Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian |
| topic | Spectral Theory Mathematical Physics Algebraic Geometry 15A29, 47B36, 81Q35 |
| url | https://arxiv.org/abs/2602.08304 |