Abstract
Abstract
The Stone theorem requires that in a physical Hilbert space $${{{\mathcal {H}}}}$$
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 $${{{\mathcal {K}}}}$$
K
in which H is nonhermitian but $${{\mathcal {PT}}}$$
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 $${{{\mathcal {H}}}}$$
H
. For a $${{\mathcal {PT}}}$$
PT
-symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of $${{{\mathcal {H}}}}$$
H
becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of $${{{\mathcal {H}}}}$$
H
enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.
Publisher
Springer Science and Business Media LLC
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