Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators

The recently introduced by us two- and three-parameter ($p,q$)- and ($p,q,\mu$)-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper we explore certain Hermitian Hamiltonian build in terms of non-Hermitian position and momentum operators obeying definite $\eta(N)$-pseudo-Hermiticity properties. A generalized nonlinear (with the coefficients depending on the excitation number operator $N$) one-mode Bogolyubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of "almost free" (but essentially nonlinear) Hamiltonian.