Floquet Isospectrality of the Zero Potential for Discrete Periodic Schr\"odinger Operators

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