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

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