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Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplusq_d\mathbb{Z}$, 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 theoperators $\Delta+V$ and $\Delta$ are Floquet isospectral. We then usecombinatorial methods to validate these numerical solutions.

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