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.

v3: 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

References in corpus (5)

• Quantum differentiability on quantum tori
• Cwikel-Solomyak estimates on tori and Euclidean spaces
• Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials
• Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature
• Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry