On the pseudo-norm and admissible solutions of the -symmetric Scarf I potential

Abstract

The physically admissible solutions of the -symmetric Scarf I potential are identified in the domain of real and complex energies. It is found that generally there are no admissible complex-energy solutions, and there is one with real energy. In a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi-parity q = ±; however, the two sets are not -orthogonal in the domain of real energies. The sign of the pseudo-norm of states with real energy is found to oscillate as (−1)n, in accordance with the example of other -symmetric potentials possessing an infinite number of discrete levels. It is argued that the spontaneous breakdown of -symmetry cannot be defined for the Scarf I potential. A comparison with some -symmetric extensions of the infinite square well is presented.