Weak separability and partial Fermi isospectrality of discrete periodic Schr\"odinger operators

TL;DR

This paper introduces new concepts of generalized partial Fermi isospectrality and weak separability for discrete periodic Schrödinger operators, showing their equivalence and implications for spectral properties and separability of potentials.

Contribution

It defines generalized partial Fermi isospectrality and weak separability, proving their equivalence and applying these notions to spectral analysis of periodic Schrödinger operators.

Findings

Generalized partial Fermi isospectral potentials share the same weak separability.

Potentials with generalized partial Fermi isospectrality have the same separability properties.

Components of generalized Fermi isospectral potentials are Floquet isospectral in some sense.

Abstract

In this paper, we consider the discrete periodic Schr\"odinger operators $\Delta+V$ on $\Z^d$, where $V$ is $\Gamma$-periodic with $\Gamma=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$, $j=1,2,\cdots,d,$ are pairwise coprime. We introduce the notions of generalized partial Fermi isospectrality and weak separability, and prove that two generalized partially Fermi isospectral potentials have the same weak separability. As a direct application, we can prove that two potentials have the same $(d_1,d_2,\cdots,d_r)$-separability by assuming that they are generalized partially Fermi isospectral, instead of the Fermi isospectrality or Floquet isospectrality. Besides, we prove that each couples of components of the generalized Fermi isospectral potentials are Floquet isospectral in some sense.