Floquet isospectrality for periodic graph operators
Institutional Repository Document
Overview
Research
Identity
View All
Overview
abstract
Let $Gamma=q_1mathbb{Z}oplus q_2 mathbb{Z}opluscdotsoplus q_dmathbb{Z}$ with arbitrary positive integers $q_l$, $l=1,2,cdots,d$. Let $Delta_{ m discrete}+V$ be the discrete Schr"odinger operator on $mathbb{Z}^d$, where $Delta_{ m discrete}$ is the discrete Laplacian on $mathbb{Z}^d$ and the function $V:mathbb{Z}^d o mathbb{C}$ is $Gamma$-periodic. We prove two rigidity theorems for discrete periodic Schr"odinger operators: (1) If real-valued $Gamma$-periodic functions $V$ and $Y$ satisfy $Delta_{ m discrete}+V$ and $Delta_{ m discrete}+Y$ are Floquet isospectral and $Y$ is separable, then $V$ is separable. (2) If complex-valued $Gamma$-periodic functions $V$ and $Y$ satisfy $Delta_{ m discrete}+V$ and $Delta_{ m discrete}+Y$ are Floquet isospectral, and both $V=\bigoplus_{j=1}^rV_j$ and $Y=\bigoplus_{j=1}^r Y_j$ 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.
