We prove bounds of the form where is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices, where we get a generalized Nevai conjecture about an -condition implying a Szegő condition. One key is a general new form of the Birman-Schwinger bound in gaps.
"Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices." Duke Math. J. 157 (3) 461 - 493, 15 April 2011. https://doi.org/10.1215/00127094-1272912