Band edges of periodic Schrödinger operators are generically isolated and nondegenerate

Abstract

For periodic Schrödinger operators $H_V=-\Delta +V$ with bounded real-valued potentials on ${\mathbb{R}}^d$ with $d\ge 2$, 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 Schrödinger operators.

16 pages, comments are welcome