Quantum phase transitions in nonhermitian harmonic oscillator

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.