On position and momentum operators in theq-oscillator

Abstract

The position and momentum operators of the q-oscillator (with the main relation aa+ − qa+a = 1) are symmetric but not self-adjoint if q > 1. They have one-parameter family of self-adjoint extensions. These extensions are given explicitly. Their spectra and eigenfunctions are derived. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a+ and a of the q-oscillator at 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.