  The Stone theorem requires that in a physical Hilbert space ${\cal H}$ the
time-evolution of a stable quantum system is unitary if and only if the
corresponding Hamiltonian $H$ is self-adjoint. Sometimes, a simpler picture of
the evolution may be constructed in a manifestly unphysical Hilbert space
${\cal K}$ in which $H$ is nonhermitian but ${\cal PT}-$symmetric. In
applications, unfortunately, one only rarely succeeds in circumventing the key
technical obstacle which lies in the necessary reconstruction of the physical
Hilbert space ${\cal H}$. For a ${\cal PT}-$symmetric version of the spiked
harmonic oscillator we show that in the dynamical regime of the unavoided level
crossings such a reconstruction of ${\cal H}$ becomes feasible and, moreover,
obtainable by non-numerical means. The general form of such a reconstruction of
${\cal H}$ enables one to render every exceptional unavoided-crossing point
tractable as a genuine, phenomenologically most appealing
quantum-phase-transition instant.
