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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 operatorsand finite gap Jacobi matrices where we get a generalized Nevai conjectureabout an $L^1$ condition implying a Szeg\H{o} condition. One key is a generalnew form of the Birman--Schwinger bound in gaps.

Critical Lieb-Thirring Bounds in Gaps and the Generalized Nevai Conjecture for Finite Gap Jacobi Matrices