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

摘要

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