[Paper Review] Operators of the $q$-oscillator

Franciszek Hugon Szafraniec|ArXiv.org|Nov 20, 2007

Mathematical Analysis and Transform Methods9 references3 citations

TL;DR

This paper investigates unbounded subnormal operators in the context of the $q$-oscillator algebra, introducing a new operator $C$ that $q$-commutes with creation and annihilation operators, enabling a unified treatment of bounded and unbounded cases. The key contribution is a characterization of subnormality via spectral measures, showing uniqueness of the representing measure for $0 < q \leq 1$, while for $q > 1$, non-unique, non-rotationally invariant measures exist, implying non-uniqueness of normal extensions.

ABSTRACT

We scrutinize the possibility of extending the result of \cite{ccr} to the case of q-deformed oscillator for $q$ real; for this we exploit the whole range of the deformation parameter as much as possible. We split the case into two depending on whether a solution of the commutation relation is bounded or not. Our {\it leitmotif} is {\it subnormality}. The deformation parameter $q$ is reshaped and this is what makes our approach effective. The newly arrived parameter, the operator $C$, has two remarkable properties: it separates in the commutation relation the annihilation and creation operators from the deformation as well as it $q$-commutes with those two. This is why introducing the operator $C$ seems to be far-reaching.

Motivation & Objective

• To extend the subnormality framework from the standard harmonic oscillator ($q=1$) to the $q$-deformed oscillator for real $q$.
• To analyze bounded vs. unbounded solutions of the $q$-oscillator commutation relation $S^*S - qSS^* = I$.
• To characterize when the weighted shift realization of the $q$-oscillator is subnormal via spectral measures.
• To resolve the issue of uniqueness of normal extensions for $q > 1$, where multiple non-unitarily equivalent extensions exist.

Proposed method

• Introduce a new operator $C$ that $q$-commutes with both creation and annihilation operators, separating deformation from the algebraic structure.
• Use the $q$-deformation of the factorial $[n]_q!$ to define a reproducing kernel Hilbert space $\mathcal{H}_q$ of analytic functions.
• Represent the creation operator as multiplication by $z$ and the annihilation operator as the $q$-derivative $D_q$, forming a weighted shift on $\mathcal{H}_q$.
• Apply spectral theory to characterize subnormality via the existence of a representing measure $\mu$ for the moments $\langle M^n 1, 1 \rangle = [n]_q!$.
• Use the moment problem framework to distinguish between determinate and indeterminate measures, particularly for $q > 1$, based on Stieltjes indeterminacy.
• Leverage rotationally invariant and non-invariant measures to show non-uniqueness of normal extensions when $q > 1$.

Experimental results

Research questions

• RQ1 Under what conditions is the $q$-oscillator creation operator $M$ (multiplication by $z$) subnormal in the Hilbert space $\mathcal{H}_q$?
• RQ2 How does the spectral measure $\mu$ associated with the $q$-oscillator depend on $q$, and when is it unique?
• RQ3 Why does the $q$-oscillator fail to have a cyclic normal extension when $q > 1$, despite being subnormal?
• RQ4 What role does the $q$-commuting operator $C$ play in unifying the analysis of bounded and unbounded cases?
• RQ5 Can non-rotationally invariant measures exist for the $q$-oscillator when $q > 1$, and what does this imply for the uniqueness of normal extensions?

Key findings

• For $0 < q < 1$, the multiplication operator $M$ on $\mathcal{H}_q$ is bounded and subnormal, and the representing measure $\mu$ is unique.
• For $q = 1$, the operator $M$ is unbounded and subnormal, and the measure $\mu$ is uniquely determined, corresponding to the standard harmonic oscillator.
• For $q > 1$, the operator $M$ is unbounded and subnormal but admits multiple non-unitarily equivalent normal extensions, implying non-uniqueness of the spectral measure.
• The existence of non-rotationally invariant measures for $q > 1$ is equivalent to the Stieltjes indeterminacy of the moment sequence $[n]_q!$, which holds due to the growth of $[n]_q!$.
• The introduction of the $q$-commuting operator $C$ allows a unified treatment of the $q$-oscillator algebra and clarifies the role of deformation in subnormality.
• Explicit examples of non-rotationally invariant measures exist for $q > 1$, confirming the failure of uniqueness in the spectral problem for the $q$-oscillator.