Floquet isospectrality for periodic graph operators

Abstract

Let Γ=q1Z⊕q2Z⊕⋯⊕qdZ with arbitrary positive integers ql, l=1,2,⋯,d. Let Δdiscrete+V be the discrete Schrödinger operator on Zd, where Δdiscrete is the discrete Laplacian on Zd and the function V:Zd→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 Δdiscrete+V and Δdiscrete+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 Δdiscrete+V and Δdiscrete+Y are Floquet isospectral, and both V=⨁j=1rVj and Y=⨁j=1rYj are separable functions, then, up to a constant, lower dimensional decompositions Vj and Yj 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.