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

Abstract

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 Lp-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) Schrödinger operators that are built out of fractional Laplace-Beltrami operators and Lp-potentials. We also conjecture a semi-classical Weyl's law for such operators.

Noncommutative Wiener–Wintner type ergodic theorems

“…Consider T d = R d /Z d , and let m denote the normalized Lebesgue measure on T d . As in[MP21] we will construct a positive Dunford-Schwartz operator on the noncommutative torus T d θ defined below. Let θ = [θ j,k ] d j,k=1 be a real skew-symmetric d × d-matrix.…”