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Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices. (English) Zbl 1229.35157

The paper discusses spectral theory of Schrödinger operators on \(L^2(\mathbb R^n)\) and Jacobi matrices. The authors prove bounds of the form
\[ \sum_{e\in I\cap \sigma_d(H)} \text{dist}( e, \sigma_e(H))^{1/2}\leq L^1\text{-norm of a perturbation}, \]
where \(I\) is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices, where they 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.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
35J10 Schrödinger operator, Schrödinger equation
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