Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators

Abstract

Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in (\mathbb{Z}^+)^d$ for each $j\in \{1,\ldots,d\}$, and denote by $Δ$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. Using Macaulay2, we first numerically find complex-valued $Γ$-periodic potentials $V:\mathbb{Z}^d\to \mathbb{C}$ such that the operators $Δ+V$ and $Δ$ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.

Figures (3)

• Figure 1: The weighted digraph $J_m$ of an $m \times m$ Jacobi matrix $M$.
• Figure 2: The refined digraph $G'$. Notice we do not include the weight $0$ self loops after specializing.
• Figure 3: (Above) $G'$ after fixing the cycles $v_1$ and $v_2$ and removing cycles $v_3$ and $v_4$. (Below) $G'$ after fixing the cycles $v_3$ and $v_4$ and removing cycles $v_1$ and $v_2$. Notice how these are the same digraphs, just relabeled; thus indeed the coefficients of $v_1v_2$ and $v_3v_4$ are the same.

Theorems & Definitions (14)

• Definition 1
• Theorem 1.1
• Remark 1
• Theorem 3.1
• Theorem 3.2
• Definition 2
• Lemma 3.3
• proof
• proof : Proof of Reduction of Theorem 3.1 to Theorem 3.2
• Theorem 4.1