Fermi isospectrality for discrete periodic Schrödinger operators

Abstract

AbstractLet , where , , are pairwise coprime. Let be the discrete Schrödinger operator, where Δ is the discrete Laplacian on and the potential is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension : If at some energy level, Fermi varieties of two real‐valued Γ‐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 Γ‐periodic potentials V and Y are Fermi isospectral and both and are separable functions, then, up to a constant, lower dimensional decompositions and are Floquet isospectral, ; If a real‐valued Γ‐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”.