Asymptotics of singular values for quantum derivatives

TL;DR

This paper establishes Weyl-type asymptotics for the singular values of quantum derivatives of functions in Sobolev spaces, linking spectral properties to classical function norms using operator algebra techniques.

Contribution

It provides a novel asymptotic analysis of quantum derivatives' spectra, connecting Sobolev space norms with operator ideal norms through $C^*$-algebraic methods.

Findings

Weyl asymptotics for quantum derivatives derived

Spectral bounds expressed via Sobolev norms

Operator ideal norms linked to classical function spaces

Abstract

We obtain Weyl type asymptotics for the quantised derivative $\dbar f$ of a function $f$ from the homgeneous Sobolev space $\dot{W}^1_d(\mathbb{R}^d)$ on $\mathbb{R}^d.$ The asymptotic coefficient $\|\nabla f\|_{L_d(\mathbb R^d)}$ is equivalent to the norm of $\dbar f$ in the principal ideal $\mathcal{L}_{d,\infty},$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\dbar f.$ Our methods are based on the $C^{\ast}$-algebraic notion of the principal symbol mapping on $\mathbb{R}^d$, as developed recently by the last two authors and collaborators.