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Communications in Mathematical Physics

, Volume 334, Issue 1, pp 473–505 | Cite as

Spectral Inequalities for Jacobi Operators and Related Sharp Lieb–Thirring Inequalities on the Continuum

  • Lukas Schimmer
Article

Abstract

In this paper we approximate a Schrödinger operator on \({L^2(\mathbb{R})}\) by Jacobi operators on \({\ell^{2}(\mathbb{Z})}\) to provide new proofs of sharp Lieb–Thirring inequalities for the powers \({\gamma = \frac{1}{2}}\) and \({\gamma = \frac{3}{2}}\). To this end we first investigate spectral inequalities for Jacobi operators. Using the commutation method, we present a new, direct proof of a sharp inequality corresponding to a Lieb–Thirring inequality for the power \({\frac{3}{2}}\) on \({\ell^2(\mathbb{Z})}\). We also introduce inequalities for higher powers of the eigenvalues as well as for matrix-valued potentials and compare our results to previously established bounds.

Keywords

Negative Eigenvalue Essential Spectrum Jacobi Matrice Darboux Transformation Jacobi Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK

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