Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices

Abstract

We prove bounds of the form ${\displaystyle \sum _{e\in I\cap {\sigma }_d(H)}\mathrm{dist}(e,{\sigma }_e(H){)}^{1/2}\le L^1-\mathrm{norm} \mathrm{of} a \mathrm{perturbation},}$ where $I$ 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 $L^1$-condition implying a Szegő condition. One key is a general new form of the Birman-Schwinger bound in gaps.