  We prove bounds of the form $\sum_{e\in I\cap\sigma_\di (H)} \dist
(e,\sigma_\e (H))^{1/2} \leq L^1$-norm of a perturbation, where $I$ is a gap.
Included are gaps in continuum one-dimensional periodic Schr\"odinger operators
and finite gap Jacobi matrices where we get a generalized Nevai conjecture
about an $L^1$ condition implying a Szeg\H{o} condition. One key is a general
new form of the Birman--Schwinger bound in gaps.
