Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators

Let $\mathrm{\Gamma }=q_1\mathbb{Z}\oplus q_2\mathbb{Z}\oplus \cdots \oplus q_d\mathbb{Z}$, with $q_j\in ({\mathbb{Z}}^{+}{)}^d$ for each $j\in \{1,\dots ,d\}$, and denote by $\mathrm{\Delta }$ the discrete Laplacian on ${\mathrm{\ell }}^2\left({\mathbb{Z}}^d\right)$. Using Macaulay2, we first numerically find complex-valued $\mathrm{\Gamma }$-periodic potentials $V:{\mathbb{Z}}^d\to \mathbb{C}$ such that the operators $\mathrm{\Delta }+V$ and $\mathrm{\Delta }$ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.