Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 047, 12 pages      arXiv:1602.07927      https://doi.org/10.3842/SIGMA.2016.047

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

Alexandre M. Gavrilik a and Ivan I. Kachurik ab
a) Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna Str., Kyiv, 03680 Ukraine
b) Khmelnytskyi National University, 11 Instytutska Str., Khmelnytskyi, 29016 Ukraine

Received February 12, 2016, in final form May 06, 2016; Published online May 12, 2016

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.

Key words: deformed Heisenberg algebra; position and momentum operators; deformed oscillators; structure function of deformation; deformation parameters; ground state energy.

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