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

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.

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Correspondence to Lukas Schimmer.

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Communicated by B. Simon

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Schimmer, L. Spectral Inequalities for Jacobi Operators and Related Sharp Lieb–Thirring Inequalities on the Continuum. Commun. Math. Phys. 334, 473–505 (2015). https://doi.org/10.1007/s00220-014-2137-3

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Keywords

  • Negative Eigenvalue
  • Essential Spectrum
  • Jacobi Matrice
  • Darboux Transformation
  • Jacobi Operator