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

Let $\mathrm{\Gamma }=q_1\mathbb{Z}\oplus q_2\mathbb{Z}$ with $q_1\in {\mathbb{Z}}_{+}$ and $q_2\in {\mathbb{Z}}_{+}$. Let $\mathrm{\Delta }+X$ be the discrete periodic Schrödinger operator on ${\mathbb{Z}}^2$, where $\mathrm{\Delta }$ is the discrete Laplacian and $X:{\mathbb{Z}}^2\to \mathbb{C}$ is $\mathrm{\Gamma }$-periodic. In this paper, we develop tools from complex analysis to study the isospectrality of discrete periodic Schrödinger operators. We prove that if two $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-periodic potential, the Fermi variety $F_{\lambda }(V)\mathrm{/}{\mathbb{Z}}^2$ is irreducible for any $\lambda \in \mathbb{C}$, which partially confirms a conjecture of Gieseker, Knörrer and Trubowitz in the early 1990s.