Admissible perturbations and false instabilities in PT -symmetric quantum systems

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 $\partial {\cal D}^{(physical)}$.