Bethe–Sommerfeld conjecture

Let $d\geq 2$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be a periodic potential such that the Schrödinger operator $H_V$ is self-adjoint. Bethe–Sommerfeld conjecture. The spectrum $\sigma(H_V)$ has finitely many gaps. In one dimension, periodic Schrödinger operators can have infinitely many open gaps, whereas in higher dimensions the conjecture predicts that only finitely many gaps remain. The statement is known under various additional regularity and structural assumptions, but its general form is the subject of the paper.