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

Let $Gamma=q_1mathbb{Z}oplus q_2 mathbb{Z} $ with $q_1in mathbb{Z}_+$ and $q_2inmathbb{Z}_+$. Let $Delta+X$ be the discrete periodic Schr"odinger operator on $mathbb{Z}^2$, where $Delta$ is the discrete Laplacian and $X:mathbb{Z}^2 o mathbb{C}$ 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 $X=X_1oplus X_2$ and $Y= Y_1oplus Y_2$ 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 $lambdain mathbb{C}$, which partially confirms a conjecture of Gieseker, Kn"{o}rrer and Trubowitz in the early 1990s.

Fermi isospectrality of discrete periodic Schrdinger operators with separable potentials on $mathbb{Z}^2$ Institutional Repository Document