  Stemming from the seminal work of Bender & Boettcher in 1998 (Phys. Rev.
Lett. vol. 80 pp. 5243-5246), there has been great interest in the study of
PT-symmetric models of quantum mechanics, where the primary focus is with the
study of non-Hermitian Hamiltonians that nevertheless produce countably
infinite sets of real-valued eigenvalues. One of the fundamental models of such
a system is governed by the Hamiltonian $H = \hat{p}^2 +
x^2(ix)^{\varepsilon}$. In their work, Bender & Boettcher proposed a WKB
methodology for the prediction of the discrete eigenvalues in the so-called
unbroken region of $\varepsilon > 0$. However, the authors noted that this
methodology fails to predict those 'broken' eigenvalues for $\varepsilon < 0$.
Here, we shall explain why the traditional WKB methodology fails, and we shall
demonstrate how eigenvalues for all relevant values of $\varepsilon$ can be
predicted using techniques in exponential asymptotics. These predictions
provide excellent agreement to exact numerical results over nearly the entire
range of values. Moreover, such techniques can be extended to a much wider
range PT-symmetric problems.
