Nonstandard Deformed Oscillators from $q$- and $p,q$-Deformations of Heisenberg Algebra

TL;DR

This paper explores new realizations of $p,q$-deformed Heisenberg algebras, introducing nonstandard deformed oscillators with unique properties, including pseudo-Hermiticity, and compares them to existing solutions.

Contribution

It presents novel solutions for realizing $p,q$-deformed Heisenberg algebras, expanding the class of deformed oscillators with distinct properties and ground state energies.

Findings

New nonstandard deformed oscillators discovered

Distinct properties including $ ext{eta}(N)$-pseudo-Hermiticity established

Differences in ground state energy from previous models

Abstract

For the two-parameter $p,q$-deformed Heisenberg algebra introduced recently and in which, instead of usual commutator of $X$ and $P$ in the l.h.s. of basic relation $[X,P] = {\rm i}\hbar$, one uses the $p,q$-commutator, we established interesting properties. Most important is the realizability of the $p,q$-deformed Heisenberg algebra by means of the appropriate deformed oscillator algebra. Another uncovered property is special extension of the usual mutual Hermitian conjugation of the creation and annihilation operators, namely the so-called $\eta(N)$-pseudo-Hermitian conjugation rule, along with the related $\eta(N)$-pseudo-Hermiticity property of the position or momentum operators. In this work, we present some new solutions of the realization problem yielding new (nonstandard) deformed oscillators, and show their inequivalence to the earlier known solution and the respective deformed oscillator algebra, in particular what concerns ground state energy.