Fermi isospectrality of discrete periodic Schr\"odinger operators with separable potentials on $\mathbb{Z}^2$

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}$ with $q_1\in \mathbb{Z}_+$ and $q_2\in\mathbb{Z}_+$. Let $\Delta+X$ be the discrete periodic Schr\"odinger operator on $\mathbb{Z}^2$, where $\Delta$ is the discrete Laplacian and $X:\mathbb{Z}^2\to \mathbb{C}$ is $\Gamma$-periodic. In this paper, we develop tools from complex analysis to study the isospectrality of discrete periodic Schr\"odinger operators. We prove that if two $\Gamma$-periodic potentials $X$ and $Y$ are Fermi isospectral and both $X=X_1\oplus X_2$ and $Y= Y_1\oplus Y_2$ are separable functions, then, up to a constant, one dimensional potentials $X_j$ and $Y_j$ are Floquet isospectral, $j=1,2$. This allows us to prove that for any non-constant separable real-valued $\Gamma$-periodic potential, the Fermi variety $F_{\lambda}(V)/\mathbb{Z}^2$ is irreducible for any $\lambda\in \mathbb{C}$, which partially confirms a conjecture of Gieseker, Kn\"{o}rrer and Trubowitz in the early 1990s.