Semiclassical Calculation of the C Operator in PT-Symmetric Quantum Mechanics

To determine the Hilbert space and inner product for a quantum theory defined by a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H$, it is necessary to construct a new time-independent observable operator called $C$. It has recently been shown that for the {\it cubic} $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H=p^2+x^2+iϵx^3$ one can obtain $\mathcal{C}$ as a perturbation expansion in powers of $ϵ$. This paper considers the more difficult case of noncubic Hamiltonians of the form $H=p^2+x^2(ix{)}^{\delta }$ ($\delta \ge 0$). For these Hamiltonians it is shown how to calculate $\mathcal{C}$ by using nonperturbative semiclassical methods.