Irreversibility principle for Bloch eigenvalues

Let $L(V)$ be the operator considered in the paper, with Bloch eigenvalues depending on $V$. A Bloch eigenvalue is said to leave the real line when it becomes nonreal as $V$ varies. Irreversibility principle. Once Bloch eigenvalues have left the real line, they never return to the real line. The source presents this as one of two conjectural principles governing the spectral changes of $L(V)$, and the supplied status marks it resolved.