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In this paper we approximate a Schr\"odinger operator on $L^2(\R)$ by Jacobioperators on $\ell^2(\Z)$ to provide new proofs of sharp Lieb-Thirringinequalities for the powers $\gamma=1/2$ and $\gamma=3/2$. To this end we firstinvestigate spectral inequalities for Jacobi operators. Using the commutationmethod we present a new, direct proof of a sharp inequality corresponding to aLieb-Thirring inequality for the power 3/2 on $\ell^2(\Z)$. We also introduceinequalities for higher powers of the eigenvalues as well as for matrix-valuedpotentials and compare our results to previously established bounds.

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