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

TL;DR

The paper investigates the algebraic structure of spectral invariants for the discrete Laplacian, focusing on 1D $q$-periodic potentials and Floquet isospectrality to the zero potential. It defines the spectral-invariant polynomials $p_k$ from $D_V(\lambda)-D_{\mathbf{0}}(\lambda)$, proves these generate an ideal $I$, and constructs a Gröbner basis $G$ with leading terms $LT(g_k)=v_k^k$, yielding an affine Hilbert polynomial $HP_{R/I}(s)=n!$ and a zero-dimensional variety with $n!$ points counting multiplicity. A symmetry-aware specialization reduces the problem to fewer variables, producing an analogous Gröbner basis $G'$ and Hilbert polynomial $HP_{R/I'}(s)=2^m m!$, along with conjectures and partial proofs about nonzero complex potentials Floquet isospectral to $\Delta$ for various periods, supported by computational experiments. The study extends to general full-rank sublattices, showing that in higher dimensions some lattices admit only trivial isospectral potentials, highlighting open classification problems for $d\ge2$ and illustrating rich algebraic-geometric structure behind Floquet isospectrality in discrete periodic graphs.

Abstract

Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in \mathbb{Z}^+$ for each $j\in \{1,\ldots,d\}$, and denote by $Δ$ 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 Gröbner basis. We also present various collections of complex $Γ$-periodic potentials $V$ that are such that $Δ$ and $Δ+ V$ are Floquet isospectral. We end with a discussion of the general setting, where the $q_i$ are taken to be vectors in $\mathbb{Z}^d$.

Figures (5)

• Figure 1: All solutions for $n=5$ up to symmetries.
• Figure 2: The nonzero singular solutions for $n=6$.
• Figure 3: All specialized $n=7$ solutions up to symmetries.
• Figure 4: All solutions for $n=9$ with two zeroes in a row up to symmetry.
• Figure 5: All solutions for $n=5$, $n=6$, $n=7$, and $n=8$.

Theorems & Definitions (37)

• Lemma 2.1
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof : Proof of Theorem \ref{['grobnerbasis']}
• Theorem 3.5