Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian

TL;DR

This paper investigates the algebraic structure of spectral invariants for the discrete Laplacian on integer lattices, providing a Gr"obner basis construction and examples of Floquet isospectral potentials, mainly in one dimension.

Contribution

It introduces a detailed algebraic analysis of spectral invariants for the discrete Laplacian, including a Gr"obner basis construction for the one-dimensional case and examples of isospectral potentials.

Findings

Constructed a Gr"obner basis for the ideal of spectral invariants in 1D.

Identified collections of complex periodic potentials with Floquet isospectrality.

Discussed the extension to higher-dimensional lattice settings.

Abstract

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in \mathbb{Z}^+$ for each $j\in \{1,\ldots,d\}$, and denote by $\Delta$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. We describe various algebraic properties of the ideal of spectral invariants for the discrete Laplacian when $d=1$, including a construction of a Gr\"obner basis. We also present various collections of complex $\Gamma$-periodic potentials $V$ that are such that $\Delta$ and $\Delta + V$ are Floquet isospectral. We end with a discussion of the general setting, where the $q_i$ are taken to be vectors in $\mathbb{Z}^d$.