Proof of geometric Borg's Theorem in arbitrary dimensions

Abstract

Let $\hat{I}”+V$ be the discrete Schrödinger operator, where $\hat{I}”$ is the discrete Laplacian on ${\mathbb{Z}}^d$ and potential $V:{\mathbb{Z}}^d\to \mathbb{C}$ is $\hat{I}“$-periodic with $\hat{I}“=q_1\mathbb{Z}\oplus q_2\mathbb{Z}\oplus \cdots \oplus q_d\mathbb{Z}$. In this study, we establish a comprehensive characterization of complex-valued $\hat{I}“$-periodic functions such that the Bloch variety of $\hat{I}”+V$ contains a graph of an entire function, in particular, we show that there are exactly $q_1{q}_2\cdots q_d$ 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 $\hat{I}”+V$ contains a graph of an entire function, which confirms the conjecture concerning the geometric version of Borg's theorem in arbitrary dimensions.

J. Eur. Math. Soc. (JEMS) to appear