Floquet isospectrality for periodic graph operators

Let I' = q1Z & REG; q2Z & REG; & BULL; & BULL; & BULL; & REG;qdZ with arbitrary positive integers ql, l = 1, 2, & BULL; & BULL; & BULL;, d. Let Adiscrete + V be the discrete Schrodinger operator on Zd, where Adiscrete is the discrete Laplacian on Zd and the function V : Zd & RARR;C is P-periodic. We prove two rigidity theorems for discrete periodic Schrodinger operators:(1) If for real-valued P-periodic functions V and Y, the operators Adiscrete + V and Adiscrete + Y are Floquet isospectral and Y is separable, then V is separable.(2) If for complex-valued P-periodic functions V and Y, the operators Adiscrete + V and Adiscrete + Y are Floquet isospectral, and both V = & REG;rj=1 Vj and Y = & REG;rj=1 Yj are separable functions, then, up to a constant, lower dimensional decompositions Vj and Yj are Floquet isospectral, j = 1, 2, & BULL; & BULL; & BULL;, 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.& COPY; 2023 Elsevier Inc. All rights reserved.