Floquet isospectrality for periodic graph operators

Abstract

Let $\Gamma =q_1\mathbb{Z}\oplus q_2\mathbb{Z}\oplus \cdots \oplus q_d\mathbb{Z}$ with arbitrary positive integers $q_l$, $l=1,2,\cdots ,d$. Let ${\Delta }_{\mathrm{discrete}}+V$ be the discrete Schrödinger operator on ${\mathbb{Z}}^d$, where ${\Delta }_{\mathrm{discrete}}$ is the discrete Laplacian on ${\mathbb{Z}}^d$ and the function $V:{\mathbb{Z}}^d\to \mathbb{C}$ is $\Gamma$-periodic. We prove two rigidity theorems for discrete periodic Schrödinger operators: (1) If real-valued $\Gamma$-periodic functions $V$ and $Y$ satisfy ${\Delta }_{\mathrm{discrete}}+V$ and ${\Delta }_{\mathrm{discrete}}+Y$ are Floquet isospectral and $Y$ is separable, then $V$ is separable. (2) If complex-valued $\Gamma$-periodic functions $V$ and $Y$ satisfy ${\Delta }_{\mathrm{discrete}}+V$ and ${\Delta }_{\mathrm{discrete}}+Y$ 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, $j=1,2,\cdots ,r$. 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 study more general lattices.