In a previous paper it was shown that a one-turning-point Wentzel-Kramers-Brillouin (WKB) approximation gives an accurate picture of the spectrum of certain non-Hermitian $\mathcal{PT}$-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential $V=igx$ and the sinusoidal potential $V=ig\sin \left(\alpha x\right)$. However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential $V=ig{x}^3$, and in particular, it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a pair of turning points. The extended method gives an extremely accurate approximation to the spectrum of $V=ig{x}^3$, and more generally it works for potentials of the form $V=ig{x}^{2N+1}$. When applied to potentials with half-integral powers of $x$, the method again works well for one sign of the coupling, namely, that for which the turning points lie on the first sheet in the lower-half plane.

• Received 29 March 2012

DOI:https://doi.org/10.1103/PhysRevA.85.052118