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arXiv · 1310.3764

Spectral inequalities for Jacobi operators and related sharp Lieb-Thirring inequalities on the continuum

Lukas Schimmer
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Abstract

In this paper we approximate a Schrödinger operator on $L^2(\R)$ by Jacobi operators on $\ell^2(\Z)$ to provide new proofs of sharp Lieb-Thirring inequalities for the powers $γ=1/2$ and $γ=3/2$. 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 3/2 on $\ell^2(\Z)$. 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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Lukas Schimmer. 2013-10-14. Spectral inequalities for Jacobi operators and related sharp Lieb-Thirring inequalities on the continuum. https://doi.org/10.1007/s00220-014-2137-3

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