$\mathcal{P}\mathcal{T}$-Symmetric Generalized Extended Momentum Operator

We develop further the concept of generalized extended momentum operator (GEMO), which has been introduced very recently in \citep{M.H2}, and propose the so called $\mathcal{P}\mathcal{T}$-symmetric GEMO. In analogy with GEMO, the $\mathcal{P}\mathcal{T}$-symmetric GEMO also satisfies the extended uncertainty principle (EUP) relation. Moreover, the corresponding Hamiltonian that is constructed upon the $\mathcal{P}\mathcal{T}$-symmetric GEMO, with a real or $\mathcal{P}\mathcal{T}$-symmetric potential, remains non-Hermitian but $\mathcal{P}\mathcal{T}$-symmetric and consequently its energy and momentum eigenvalues are real. We apply our formalism to a quasi-free quantum particle and the exact solutions for the energy spectrum are presented.