Proof of geometric Borg's Theorem in arbitrary dimensions
Wencai Liu
TL;DR
The paper geometrizes Borg’s theorem for discrete periodic Schrödinger operators in arbitrary dimensions by characterizing when the Bloch variety contains a graph of an entire function. It proves there are exactly $Q=\prod_{j=1}^d q_j$ complex-valued $Γ$-periodic potentials (up to Floquet isospectrality and translation) with this property, and that for real potentials the graph-embedding occurs if and only if the potential is constant, thereby establishing the geometric Borg theorem in any dimension. The approach fuses Laurent-polynomial representations of the Bloch determinant, multivariable perturbation theory to obtain holomorphic eigenvalue branches, and algebraic-geometry arguments to count and realize solutions, linking spectral rigidity with Floquet-isospectral families and offering a precise finite classification. This advances understanding of inverse spectral problems on periodic graphs by connecting Bloch/Fermi varieties with rigidity phenomena and providing explicit realizability results.
Abstract
Let $Δ+V$ be the discrete Schrödinger operator, where $Δ$ is the discrete Laplacian on $\mathbb{Z}^d$ and potential $V:\mathbb{Z}^d\to \mathbb{C}$ is $Γ$-periodic with $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$. In this study, we establish a comprehensive characterization of complex-valued $Γ$-periodic functions such that the Bloch variety of $Δ+V$ contains a graph of an entire function, in particular, we show that there are exactly $q_1q_2\cdots q_d$ such functions (up to Floquet isospectrality and translation). Moreover, by applying this understanding to real-valued functions $V$, we prove that $V$ is constant if and only if the Bloch variety of $Δ+V$ contains a graph of an entire function, which confirms the conjecture concerning the geometric version of Borg's theorem in arbitrary dimensions.