The mergings of energy levels associated with the breaking of symmetry in the model of Bender and Boettcher, and in its generalization to incorporate a centrifugal term, are analysed in detail. Even though conventional WKB techniques fail, it is shown how the ODE/IM correspondence can be used to obtain a systematic approximation scheme which captures all previously observed features. Nonperturbative effects turn out to play a crucial role, governing the behaviour of almost all levels once the symmetry-breaking transition has been passed. In addition, a novel treatment of the radial Schrödinger equation is used to recover the values of local and non-local conserved charges in the related integrable quantum field theories, without any need for resummation even when the angular momentum is nonzero.
Dorey, P., Millican-Slater, A., & Tateo, R. (2005). Beyond the WKB approximation in PT-symmetric quantum mechanics. Journal of Physics A: Mathematical and General, 38(6), 1305-1332. https://doi.org/10.1088/0305-4470/38/6/010