A Resolvent Approach to the Real Quantum Plane

TL;DR

This paper investigates the operator relation $AB=qBA$ for self-adjoint operators on a Hilbert space, providing a resolvent-based analysis and characterizing two classes of well-behaved representations.

Contribution

It introduces a resolvent approach to analyze the relation $AB=qBA$ and characterizes two classes of representations in detail.

Findings

Characterization of two classes of well-behaved representations

Development of resolvent equations for the operator relation

Insight into the structure of the quantum plane operators

Abstract

Let $q\neq \pm 1$ be a complex number of modulus one. This paper deals with the operator relation $AB=qBA$ for self-adjoint operators $A$ and $B$ on a Hilbert space. Two classes of well-behaved representations of this relation are studied in detail and characterized by resolvent equations.