Floquet isospectrality for periodic graph operators.

Let Γ = q 1 Z ⊕ q 2 Z ⊕ ⋯ ⊕ q d Z with arbitrary positive integers q l , l = 1 , 2 , ⋯ , d. Let Δ discrete + V be the discrete Schrödinger operator on Z d , where Δ discrete is the discrete Laplacian on Z d and the function V : Z d → C is Γ-periodic. We prove two rigidity theorems for discrete periodic Schrödinger operators: (1) If for real-valued Γ-periodic functions V and Y , the operators Δ discrete + V and Δ discrete + Y are Floquet isospectral and Y is separable, then V is separable. (2) If for complex-valued Γ-periodic functions V and Y , the operators Δ discrete + V and Δ discrete + Y are Floquet isospectral, and both V = ⨁ j = 1 r V j and Y = ⨁ 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 , ⋯ , 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 studying more general lattices. [ABSTRACT FROM AUTHOR]