Spectral Asymptotics for Quantized Derivatives on Quantum Euclidean Spaces

TL;DR

This paper establishes spectral asymptotics for quantized derivatives on quantum Euclidean spaces, extending previous results using advanced noncommutative analysis techniques.

Contribution

It introduces a novel approach combining noncommutative Wiener-Ikehara theorem and $C^*$-algebraic pseudo-differential operator theory to analyze spectral properties.

Findings

Spectral asymptotics are derived for quantized derivatives.

Extension of earlier results to quantum Euclidean spaces.

Application of noncommutative Tauberian theorems in spectral analysis.

Abstract

We obtain spectral asymptotics for the quantized derivatives of elements from the first-order homogeneous Sobolev space on the quantum Euclidean space, extending an earlier result of McDonald, Sukochev and Xiong (Commun. Math. Phys. 2020). Our approach is based on a noncommutative Wiener-Ikehara Tauberian theorem and a recently developed $C^\ast$-algebraic version of pseudo-differential operator theory.