New version of pseudo-hermiticity in the two-sided deformation of Heisenberg algebra

The recently introduced two- and three-parameter ($p,q$)- and ($p,q,\mu$)-deformed extensions of the Heisenberg algebra were explored under the condition of their connectedness with the respective nonstandard (other than known ones) deformed quantum oscillator algebras. In this paper we show that such connection dictates certain new $\eta (N)$-pseudo-Hermitian conjugation rule between the creation and annihilation operators, with $\eta (N)$ depending on the particle number operator $N$. In turn, that leads to the related $\eta (N)$-pseudo-Hermiticity of the position/momentum operators, though the involved Hamiltonian is Hermitian. Different possible cases are studied, and some interesting features implied by the use of such $\eta (N)$-based conjugation rule are emphasized.