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

Abstract

We prove bounds of the form ∑_(e∈I⋂σ_d(H)) dist(e, σ_e(H)^(1/2) ≤ L^1 -norm of a 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.

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