Band edges of periodic Schr\"odinger operators are generically isolated and nondegenerate

Abstract

For periodic Schr\"odinger operators $H_V=-\Delta+V$ with bounded real-valued potentials on $\mathbb R^d$ with $d\ge2$, we show that for generic potentials, each endpoint of every spectral gap is attained by a single Bloch band at only finitely many quasimomenta, and has a nondegenerate Hessian at every attaining point. This proves the Spectral Edge Conjecture for periodic Schr\"odinger operators.