Fermi isospectrality for discrete periodic Schrodinger operators

Let Gamma=q1Z OPLUS;q2Z OPLUS; horizontal ellipsis & OPLUS;qdZ$\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$, where ql & ISIN;Z+$q_l\in \mathbb {Z}_+$, l=1,2, horizontal ellipsis ,d$l=1,2,\ldots ,d$, are pairwise coprime. Let & UDelta;+V$\Delta +V$ be the discrete Schrodinger operator, where & UDelta; is the discrete Laplacian on Zd$\mathbb {Z}<^>d$ and the potential V:Zd & RARR;C$V:\mathbb {Z}<^>d\rightarrow \mathbb {C}$ is & UGamma;-periodic. We prove three rigidity theorems for discrete periodic Schrodinger operators in any dimension d & GE;3$d\ge 3$: If at some energy level, Fermi varieties of two real-valued & UGamma;-periodic potentials V and Y are the same (this feature is referred to as Fermi isospectrality of V and Y), and Y is a separable function, then V is separable;If two complex-valued & UGamma;-periodic potentials V and Y are Fermi isospectral and both V=⨁j=1rVj$V=\bigoplus _{j=1}<^>rV_j$ and Y=⨁j=1rYj$Y=\bigoplus _{j=1}<^>r Y_j$ are separable functions, then, up to a constant, lower dimensional decompositions Vj$V_j$ and Yj$Y_j$ are Floquet isospectral, j=1,2, horizontal ellipsis ,r$j=1,2,\ldots ,r$;If a real-valued & UGamma;-potential V and the zero potential are Fermi isospectral, then V is zero.(1)(2)(3)In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption "Fermi isospectrality" with a stronger assumption "Floquet isospectrality".