Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian

Let $\mathrm{\Gamma }=q_1\mathbb{Z}\oplus q_2\mathbb{Z}\oplus \cdots \oplus q_d\mathbb{Z}$, with $q_j\in {\mathbb{Z}}^{+}$ for each $j\in \{1,\dots ,d\}$, and denote by $\mathrm{\Delta }$ the discrete Laplacian on ${\mathrm{\ell }}^2\left({\mathbb{Z}}^d\right)$. 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 $\mathrm{\Gamma }$-periodic potentials $V$ that are such that $\mathrm{\Delta }$ and $\mathrm{\Delta }+V$ are Floquet isospectral. We end with a discussion of the general setting, where the $q_i$ are taken to be vectors in ${\mathbb{Z}}^d$.