Asymptotics of singular values for quantised derivatives on noncommutative tori

Abstract

We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus Tθd of the form TIθ−1. Here T stands for classical pseudodifferential operators on quantum torus of order 0, and Iθ−1 is the Riesz potential of order −1. As an application, we obtain Weyl type asymptotics for the quantised derivative ▪ of an operator x from the homogeneous Sobolev space W˙d1(Tθd). The asymptotic coefficient is equivalent to the norm of ▪ in the principal ideal Ld,∞, as well as the norm of x in W˙d1(Tθd). We precise and rectify earlier results in our previous paper (Comm. Math. Phys. 371 (2019), no. 3, 1231–1260) on quantum integration formula for ▪.