Proof of geometric Borg's Theorem in arbitrary dimensions
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Let $Delta+V$ be the discrete Schr"odinger operator, where $Delta$ is the discrete Laplacian on $mathbb{Z}^d$ and the potential $V:mathbb{Z}^d o mathbb{C}$ is $Gamma$-periodic with $Gamma=q_1mathbb{Z}oplus q_2 mathbb{Z}opluscdotsoplus q_dmathbb{Z}$. In this study, we establish a comprehensive characterization of the complex-valued $Gamma$-periodic functions such that the Bloch variety of $Delta+V$ contains the graph of an entire function, in particular, we show that there are exactly $q_1q_2cdots q_d$ such functions (up to the Floquet isospectrality and the translation). Moreover, by applying this understanding to real-valued functions $V$, we confirm the conjecture concerning the geometric version of Borg's theorem in arbitrary dimensions.