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


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.


Negative Eigenvalue Essential Spectrum Jacobi Matrice Darboux Transformation Jacobi Operator 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK

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