2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/109309$PT$ symmetric quantum mechanics for a particle trapped by the generalized non-Hermitian harmonic oscillator potential is studied. It is shown that energy and the expectation value of the position operator $x$ can not be real simultaneously, if the particle is trapped. Non-vanishing boundary conditions for the trapped particle in $PT$ symmetric theory are also discussed.10 pagesQuantum Physicstext