It is necessary to calculate the $\mathcal{C}$ operator for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H=\frac{1}{2}{p}^2+\frac{1}{2}{\mu }^2{x}^2-\lambda x^4$ in order to demonstrate that $H$ defines a consistent unitary theory of quantum mechanics. However, the $\mathcal{C}$ operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian $H=\frac{1}{2}{p}^2+\frac{1}{2}{\mu }^2{x}^2+ig{x}^3-\lambda x^4$, whose $\mathcal{C}$ operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for $\mathcal{C}$ having a nontrivial dependence on the dynamical variables $x$ and $p$ and on the parameter $\lambda$.

• Received 6 September 2005

DOI:https://doi.org/10.1103/PhysRevD.73.025002