  In ${\cal PT}-$symmetric quantum mechanics one of the most characteristic
mathematical features of the formalism is the explicit Hamiltonian-dependence
of the physical Hilbert space of states ${\cal H}={\cal H}(H)$. Some of the
most important physical consequences are discussed, with emphasis on the
dynamical regime in which the system is close to the quantum phase transition.
Consistent perturbation treatment of such a regime is proposed. An illustrative
application of the innovated perturbation theory to a non-Hermitian but ${\cal
PT}-$symmetric user-friendly family of $J-$parametric "discrete anharmonic"
quantum Hamiltonians $H=H(\vec{\lambda})$ is given. The models are shown to
admit the standard probabilistic interpretation if and only if the parameters
remain compatible with the reality of the spectrum, $\vec{\lambda} \in {\cal
D}^{(physical)}$. In contradiction to the conventional wisdom the systems are
shown stable with respect to the admissible perturbations lying inside the
domain ${\cal D}^{(physical)}$. This observation holds even in the immediate
vicinity of the phase-transition boundaries $\partial {\cal D}^{(physical)}$.
