Spontaneous breakdown of $\mathcal{P}\mathcal{T}$ symmetry in the complex Coulomb potential

Abstract

The $\mathcal{P}\mathcal{T}$ symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $\mathcal{P}\mathcal{T}$ symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $\mathcal{P}\mathcal{T}$ -symmetric if the L parameter that corresponds to angular momentum in the Hermitian case is real. $\mathcal{P}\mathcal{T}$ symmetry is spontaneously broken, however, for complex L values of the form L = −1/2 + iλ. In this case the potential remains $\mathcal{P}\mathcal{T}$ -symmetric, while the two independent solutions are transformed to each other by the $\mathcal{P}\mathcal{T}$ operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.