  In this paper, we establish Cwikel-type estimates for noncommutative tori for
any dimension~$n\geq 2$. We use them to derive a Cwikel-Lieb-Rozenblum
inequality for the number of negative eigenvalues of fractional Schr\"{o}dinger
operators on noncommutative tori in any dimension~$n\geq 2$. In order to do so
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.
