Choice of a metric for the non-Hermitian oscillator
The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian PT -symmetric part, can be transformed into a Hermitian Hamiltonian. This is achieved by introducing a metric which, in general, renders other observables such as the usual momentum or position as non-Hermitian operators. The metric depends on one real parameter, the full range of which i investigated. The explicit functional dependence of the metric and each asociated Hamiltonian is given. Aspecific choice of this parameter determines a specific combination of position and momentum as being an observable; this can be in particular either standard position or momentum, but not both simultaneously. Singularities of the metric are explored and their removability is investigated. The physical significance of these findings is discussed. © 2007 IOP Publishing Ltd.