Irreducibility of the Bloch variety for periodic Schr\"odinger operators in arbitrary dimension

Abstract

Let $d\ge1$ and $V\in C(\mathbb{T}^d;\mathbb{C})$. We prove that the Bloch variety of $-\Delta+V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spectral band functions with integer translations of quasimomentum. Using analytic geometry and perturbation estimates along imaginary rays, we show that, after suitable integer translations, all irreducible components contain the same holomorphic spectral branch.