Fermi Isospectrality of Discrete Periodic Schrödinger Operators with Separable Potentials on $$\mathbb {Z}^2$$

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