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

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 $\mathbb{Z}^d$.

💡 Research Summary

The paper investigates the algebraic structure underlying Floquet isospectrality for the discrete Laplacian on a periodic lattice. The authors focus on the one‑dimensional case where the period lattice is Γ = nℤ (n ≥ 3) and consider Γ‑periodic complex potentials V = (v₁,…,v_n). The discrete Laplacian Δ acts on ℓ²(ℤ) and the perturbed operator Δ+V is a discrete periodic Schrödinger operator. By imposing Floquet–Bloch boundary conditions, the spectral problem reduces to studying the finite Floquet matrix L_V(z). Evaluating at z = 1 yields a tridiagonal matrix whose characteristic polynomial D_V(λ) encodes the spectrum.

Floquet isospectrality of Δ and Δ+V is equivalent to the equality D_V(λ) = D_0(λ). Comparing coefficients of λ^{n−k} leads to a system of n polynomial equations p_k(v₁,…,v_n)=0 (k = 1,…,n). These are called the spectral invariants. The leading homogeneous part of p_k is the elementary symmetric polynomial e_k; lower‑degree terms f_k have degree < k−1. The invariants are invariant under the dihedral group D_n (rotations and reflections of the index set) and exhibit parity constraints: if n−k is odd, p_k contains only odd‑degree monomials, otherwise only even‑degree monomials.

The ideal I = ⟨p₁,…,p_n⟩ generated by the spectral invariants is the central algebraic object. To describe I explicitly, the authors introduce complete homogeneous symmetric polynomials H(a,b) = Σ_{a≤j₁≤…≤j_b≤n} v_{j₁}…v_{j_b} and define new polynomials

 g_k = –k Σ_{j=1}^k H(k, k−j)(–1)^j p_j  (k = 1,…,n).

Theorem 3.1 proves that G = {g₁,…,g_n} forms a Gröbner basis of I with respect to the graded reverse lexicographic (grevlex) order. The proof relies on showing that the leading term of g_k is v_k^k (Corollary 3.4) and that the S‑polynomials reduce to zero, satisfying Buchberger’s criterion. Moreover, Theorem 3.3 shows that the highest‑degree homogeneous part of g_k coincides with H(k,k), confirming that the Gröbner basis respects the symmetric structure.

From the Gröbner basis the authors compute the affine Hilbert polynomial of the quotient ring R/I (R = ℂ