Hilbert spaces of states of $\,{\cal PT}-$symmetric harmonic oscillator near exceptional points

Although the Stone theorem requires that in a physical Hilbert space ${\cal H}$ the time-evolution of a quantum system is unitary if and only if the corresponding Hamiltonian $H$ is self-adjoint in ${\cal H}$, an equivalent, recently popular picture of the same evolution may be constructed in another, manifestly unphysical Hilbert space ${\cal K}$ in which $H$ is non-Hermitian but ${\cal PT}-$symmetric. Unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary ultimate reconstruction of all of the eligible physical Hilbert spaces of states ${\cal H}$ in which $H$ is self-adjoint. We show that, how, and why such a reconstruction becomes feasible for a spiked harmonic oscillator in a phenomenologically most interesting vicinity of its phase-transition exceptional points.