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.
Categories: math-ph math.AG math.CV math.MP math.SP