Fermi isospectrality for discrete periodic Schrodinger operators

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, where $q_l\in \mathbb{Z}_+$, $l=1,2,\cdots,d$. Let $\Delta+V$ be the discrete Schr\"odinger operator, where $\Delta$ is the discrete Laplacian on $\mathbb{Z}^d$ and the potential $V:\mathbb{Z}^d\to \mathbb{R}$ is $\Gamma$-periodic. We prove three rigidity theorems for discrete periodic Schr\"odinger operators in any dimension $d\geq 3$: (1) if at some energy level, Fermi varieties of the $\Gamma$-periodic potential $V$ and the $\Gamma$-periodic potential $Y$ are the same (this feature is referred to as {\it Fermi isospectrality} of $V$ and $Y$), and $Y$ is a separable function, then $V$ is separable; (2) if potentials $V$ and $Y$ are Fermi isospectral and both $V=\bigoplus_{j=1}^rV_j$ and $Y=\bigoplus_{j=1}^r Y_j$ are separable functions, then, up to a constant, lower dimensional decompositions $V_j$ and $Y_j$ are Floquet isospectral, $j=1,2,\cdots,r$; (3) if a potential $V$ and the zero potential are Fermi isospectral, then $V$ is zero. In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption "Fermi isospectrality" with a stronger assumption "Floquet isospectrality".