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 $\gamma =1/2$ and $\gamma =3/2$. 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 ${\ell }^2(\mathbb{Z})$. 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.

28 pages, 1 figure

References in corpus (1)

• On some sharp spectral inequalities for Schrödinger operators on semi-axis