Complex WKB analysis of a PT-symmetric eigenvalue problem

The spectra of a particular class of -symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper, we present an explanation for these spectral properties in terms of quantization conditions obtained from the complex WKB method. In particular, we consider the relation of the quantization conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al (2001 J. Phys. A: Math. Gen. 34 5679 (Preprint hep-th/0103051), 2001 J. Phys. A: Math. Gen. 34 L391 (Preprint hep-th/0104119)). © 2007 IOP Publishing Ltd.