Spontaneous breakdown of PT symmetry

in the complex Coulomb potential

G LÉVAI

Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI),

P.O. Box 51, H-4001 Debrecen, Hungary

E-mail: levai@namafia.atomki.hu

Abstract. The PT symmetry of the Coulomb potential and its

solutions are studied along trajectories satisfying the PT

symmetry requirement. It is shown that with appropriate

normalization constant the general solutions can be chosen

PT-symmetric if the $L$ parameter that corresponds to angular

momentum in the Hermitian case is real. PT symmetry is

spontaneously broken, however, for complex $L$ values of the form

$L=-\frac{1}{2}+{\rm i}\lambda$. In this case the potential remains

PT-symmetric, while the two independent solutions are

transformed to each other by the PT operation and at the

same time, the two series of discrete energy eigenvalues turn into

each other's complex conjugate.

Keywords. Spontaneous breakdown of PT symmetry; Coulomb

potential; complex energy eigenvalues.

PACS Nos 03.65.Ge; 03.65.Nk; 11.30.Er