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In a previous paper we established Cwikel-type estimates on noncommutativetori and used them to get analogues in this setting of theCwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negativeeigenvalues of fractional Schr\"odinger operators. In this paper, we focus on"curved" NC tori, where the role of the usual Laplacian is played byLaplace-Beltrami operators associated with arbitrary Riemannian metrics. TheCwikel-type estimates of our previous paper are extended to pseudodifferentialoperators and powers of Laplace-Beltrami operators. There are severalapplications of these estimates. First, we get $L_p$-versions of the usualformula for the trace of \psidos\ on NC tori, i.e., for combinations of\psidos\ with $L_p$-position operators. Next, we get $L_p$-versions of theanalogues for NC tori Connes' trace theorem and Connes' integration formula.They give formulas for the NC integrals (a.k.a.\ Dixmier traces) of products of$L_p$-position operators with \psidos\ or powers of the Laplace-Beltramioperators. Moreover, by combining our Cwikel-type estimates with suitableversions of the Birman-Schwinger principle we get versions of the CLR andLieb-Thirring inequalities for negative eigenvalues of fractional Schr\"odingeroperators associated with powers of Laplace-Beltrami operators and$L_p$-potentials. As in the original Euclidean case the Lieb-Thirringinequalities imply a dual Sobolev inequality for orthonormal families. Finally,we discuss spectral asymptotics and semiclassical Weyl's laws for the ourclasses of operators on curved NC tori. This superseded a previous conjecturein our previous paper.

Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori