PT-symmetric eigenvalues for homogeneous potentials

We consider one-dimensional Schrödinger equations with homogeneous potential, under appropriate PT-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as the degree of the potential 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 as they tend to infinity.

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