Floquet isospectrality for periodic graph operators

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ with arbitrary positive integers $q_l$, $l=1,2,\cdots,d$. Let $\Delta_{\rm discrete}+V$ be the discrete Schr\"odinger operator on $\mathbb{Z}^d$, where $\Delta_{\rm discrete}$ is the discrete Laplacian on $\mathbb{Z}^d$ and the function $V:\mathbb{Z}^d\to \mathbb{C}$ is $\Gamma$-periodic. We prove two rigidity theorems for discrete periodic Schr\"odinger operators: (1) If real-valued $\Gamma$-periodic functions $V$ and $Y$ satisfy $\Delta_{\rm discrete}+V$ and $\Delta_{\rm discrete}+Y$ are Floquet isospectral and $Y$ is separable, then $V$ is separable. (2) If complex-valued $\Gamma$-periodic functions $V$ and $Y$ satisfy $\Delta_{\rm discrete}+V$ and $\Delta_{\rm discrete}+Y$ are Floquet 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, $j=1,2,\cdots,r$. Our theorems extend the results of Kappeler. Our approach is developed from the author's recent work on Fermi isospectrality and can be applied to study more general lattices.