Proof of geometric Borg’s theorem in arbitrary dimensions

Let Δ+V be the discrete Schrödinger operator, where Δ is the discrete Laplacian on Zᵈ and the potential V:Zᵈ→ C is Γ - periodic with Γ=q₁Z⊕ ⋯⊕ qdZ . We establish a comprehensive characterization of complex-valued Γ -periodic functions such that the Bloch variety of Δ+V contains the graph of an entire function; in particular, we show that there are exactly q₁⋯ qd such functions (up to Floquet isospectrality and translation). Moreover, by applying this to real-valued functions V , we prove that V is constant if and only if the Bloch variety of Δ+V contains the graph of an entire function, which confirms the conjecture concerning the geometric version of Borg’s theorem in arbitrary dimensions.