Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori

Abstract

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension~$n\ge 2$. We use them to derive Cwikel-Lieb-Rozenblum inequalities and and Lieb-Thirring inequalities for the number of negative eigenvalues of fractional Schroedinger operators on noncommutative tori in any dimension~$n\ge 2$. The latter leads to a Sobolev inequality for noncommutative tori. On the way we establish a "borderline version" of the abstract Birman-Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.

arXiv commentv3: major changes. The new version contains new material on Lieb-Thirring inequalities and Sobolev inequality on NC tori + explicit bounds for the best constants for the various inequalities established in the paper. 38 pages

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