On pseudo-Hermitian Hamiltonians and their Hermitian counterparts

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V = igx(3). In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P.