Accidental crossing of energy eigenvalues in PT-symmetric Natanzon-class potentials

The accidental crossing of energy levels is studied for a number of exactly solvable PT-symmetric potentials in one spatial dimension. This phenomenon occurs when the potential possesses two series of bound-state levels discriminated by the q=± quasi-parity quantum number and a potential parameter is tuned to specific values. In contrast with the coalescing of two such real-energy levels with the same n quantum number and continuing as a complex conjugate pair, corresponding to the breakdown of PT symmetry, accidental crossing occurs for energy levels with different n and q. In this case the energy eigenvalues become degenerate, and the corresponding wave functions become linearly dependent. It is shown that besides the known examples, the PT-symmetric harmonic oscillator, Coulomb and Scarf II potentials, this phenomenon occurs for any member of the Natanzon potential class for which the q quantum number can be defined. Two such potentials are discussed as concrete examples: the PT-symmetric generalized Ginocchio potential and a four-parameter subset of the Natanzon potential class. These potentials have been described in detail previously, however, the accidental crossing of their energy eigenvalues has not been noticed then.