Fermi isospectrality for discrete periodic Schrodinger operators
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Let $Gamma=q_1mathbb{Z}oplus q_2 mathbb{Z}opluscdotsoplus q_dmathbb{Z}$, where $q_lin mathbb{Z}_+$, $l=1,2,cdots,d$. Let $Delta+V$ be the discrete Schr"odinger operator, where $Delta$ is the discrete Laplacian on $mathbb{Z}^d$ and the potential $V:mathbb{Z}^d o mathbb{R}$ is $Gamma$-periodic. We prove three rigidity theorems for discrete periodic Schr"odinger operators in any dimension $dgeq 3$: (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".