  In a previous paper we established Cwikel-type estimates and the CLR
inequality for noncommutative tori. In this follow-up paper we extend these
results to pseudodifferential operators and to curved noncommutative tori,
where the role of the usual Laplacian is played by Laplace-Beltrami operators
associated with arbitrary densities and Riemannian metrics. The Cwikel
estimates are used to get several $L_2$ and $L_{1^+}$ Dixmier trace formulas.
Here $L_{1^+}$ is meant as the intersection of all $L_p$-spaces with $p>1$.
This extends previous Dixmier trace formulas to the $L_2$ and $L_{1^+}$
settings. Combining the Cwikel estimates with a borderline version of the
Birman-Schwinger principle leads to a CLR-type inequality for the number of
negative eigenvalues of (fractional) Schr\"odinger operators that are built out
of fractional Laplace-Beltrami operators and $L_p$-potentials. We also
conjecture a semi-classical Weyl's law for such operators.
