Real discrete spectrum of complex PT-symmetric scattering potentials

TL;DR

This paper studies how the real discrete spectrum of certain complex PT-symmetric scattering potentials evolves with parameter changes, revealing eigenvalue coalescence at exceptional points and contrasting with previous findings of eigenvalue crossings.

Contribution

It provides a detailed analysis of eigenvalue behavior in complex PT-symmetric scattering potentials, highlighting the absence of eigenvalue crossings and the occurrence of coalescences at exceptional points.

Findings

Eigenvalues coalesce at exceptional points.

No accidental real-to-real eigenvalue crossings observed.

Eigenvalue coalescence linked to finite potential barriers.

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 to the complex conjugate pairs at a finite number of exceptional points. 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.