PT-symmetric eigenvalues for homogeneous potentials

We consider one-dimensional Schrödinger equations with potential x2M(ix)ε, where M ≥ 1 is an integer and ε is real, under appropriate parity and time (PT)-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as ε changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues E as E → ∞.