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Nonstandard Deformed Oscillators from $q$- and $p,q$-Deformations of Heisenberg Algebra
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $[X,P] = \mathrm{i}\hbar$, 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.
Keywords: deformed Heisenberg algebra; position and momentum operators; deformed oscillators; structure function of deformation; deformation parameters; ground state energy. 
@article{SIGMA_2016_12_a46,
     author = {Alexandre M. Gavrilik and Ivan I. Kachurik},
     title = {Nonstandard {Deformed} {Oscillators} from $q$- and $p,q${-Deformations} of {Heisenberg} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a46/}
}
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Alexandre M. Gavrilik; Ivan I. Kachurik. Nonstandard Deformed Oscillators from $q$- and $p,q$-Deformations of Heisenberg Algebra. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a46/

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