Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension~$n\geq 2$. 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~$n\geq 2$. 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.