Spectra of observables in the q-oscillator and q-analogue of the Fourier transform

TL;DR

This paper analyzes the spectra of position and momentum operators in the q-oscillator for q>1, revealing the necessity of choosing specific self-adjoint extensions to define a physical system.

Contribution

It explicitly derives the self-adjoint extensions of the q-oscillator's operators and characterizes their spectra and eigenfunctions.

Findings

Operators are symmetric but not self-adjoint for q>1

Self-adjoint extensions are explicitly constructed

Different extensions have non-intersecting spectra

Abstract

Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa^+-qa^+a=1) are studied when q>1. These operators are symmetric but not self-adjoint. They have a one-parameter family of self-adjoint extensions. These extensions are derived explicitly. Their spectra and eigenfunctions are given. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a^+ and a of the q-oscillator for q>1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.