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Spontaneous breakdown of $\mathcal{PT}$ symmetry; Coulomb potential; complex energy eigenvalues.

The $\mathcal{PT}$ symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $\mathcal{PT}$ symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $\mathcal{PT}$ -symmetric if the 𝐿 parameter that corresponds to angular momentum in the Hermitian case is real. $\mathcal{PT}$ symmetry is spontaneously broken, however, for complex 𝐿 values of the form $L = − \dfrac{1}{2} + i\lambda$. In this case the potential remains $\mathcal{PT}$ -symmetric, while the two independent solutions are transformed to each other by the $\mathcal{PT}$ operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.

G Lévai¹

1. Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), P.O. Box 51, H-4001 Debrecen, Hungary

Published

August 2009