Real Discrete Spectrum of Complex PT-Symmetric Scattering Potentials

Abstract

We investigate the parametric evolution of the real discrete spectrum of several complex PT symmetric scattering potentials of the type $V(x)=-V_1 F_e(x) + i V_2 F_o(x), V_1>0, F_e(x)>0$ by varying $V_2$ slowly. Here e, o stand for even and odd parity and $F_{e,o}(\pm \infty )=0$. Unlike the case of Scarf II potential, we find a general absence of the recently explored accidental (real to real) crossings of eigenvalues in these scattering potentials. On the other hand, we find a general presence of coalescing of real pairs of eigenvalues at a finite number of exceptional points. After these points, real discrete eigenvalues become complex conjugate pairs. We attribute such coalescings of eigenvalues to the presence of a finite barrier (on the either side of $x=0$) which has been linked to a recent study of stokes phenomenon in the complex PT-symmetric potentials.