Petr Siegl, David Krejcirik
We show that the PT-symmetric imaginary cubic oscillator is not quasi-Hermitian. In other words, there is no quantum-mechanical Hamiltonian associated with it via physically relevant similarity transformations. Consequently, the commonly accepted quantum-mechanical interpretation of the model in terms of the modification of the inner product in the Hilbert space by means of the metric or C-operator must be completely revisited. Moreover, the non-existence of the metric operator results in the appearance of spectral instability regions. Our proof is based on the analysis of semiclassical states due to Davies, which involves a direct construction of a continuous family of approximate eigenstates of complex energies far from the spectrum. The method can be used to investigate the time evolution of fairly general initial states by expanding these in terms of the modes.
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http://arxiv.org/abs/1208.1866
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