  The mergings of energy levels associated with the breaking of PT symmetry in
the model of Bender and Boettcher, and in its generalisation 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 Schrodinger 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.
