Scarcity of real discrete eigenvalues in non-analytic complex

PT-symmetric potentials

ZAFAR AHMED

Nuclear Physics Division, Bhabha Atomic Research Centre,

Mumbai 400 085, India

E-mail: zahmed@barc.gov.in

Abstract. We find that a non-differentiability occurring whether in

real or imaginary part of a complex PT-symmetric potential

causes a scarcity of the real discrete eigenvalues despite the real

part alone possessing an infinite spectrum. We demonstrate this by

perturbing the real potentials $x^2$ and $|x|$ by imaginary

PT-symmetric potentials $ix|x|$ and $ix$, respectively.

Keywords. Non-Hermitian Hamiltonians; real discrete spectrum,

PT symmetry; complex PT-symmetric potentials; Weber

function; Airy function.

PACS Nos 03.65.Ge; 11.30.Er