Three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials

For complex PT-symmetric scattering potentials (CPTSSPs) $V(x)= V_1 f_{even}(x) + iV_2 f_{odd}(x), f_{even}(\pm \infty) = 0 = f_{odd}(\pm \infty), V_1,V_2 \in \Re $,weshowthatcomplex$k$-polesoftransmissionamplitude$t(k)$orzerosof$1/t(k)$ofthetype$\pm k_1+ik_2, k_2\ge 0$ are physical which yield three types of discrete energy eigenvalues of the potential. These discrete energies are real negative, complex conjugate pair(s) of eigenvalues (CCPEs: ${\mathcal{E}}_n\pm i{\gamma }_n$) and real positive energy called spectral singularity (SS) at $E=E_{\ast }$ where the transmission and reflection co-efficient of $V(x)$ become infinite for a special critical value of $V_2=V_{\ast }$. Based on four analytically solvable and other numerically solved models, we conjecture that a parametrically fixed CPTSSP has at most one SS. When $V_1$ is fixed and $V_2$ is varied there may exist Kato’s exceptional point(s) $(V_{EP})$ and critical values $V_{\ast m},m=0,1,2,\mathrm{.}\mathrm{.}$, so when $V_2$ crosses one of these special values a new CCPE is created. When $V_2$ equals a critical value $V_{\ast m}$ there exist one SS at $E=E_{\ast }$ along with $m$ or more number of CCPEs. Hence, this single positive energy $E_{\ast }$ is the upper (or rough upper) bound to the CCPEs: ${\mathcal{E}}_l⪅E_{\ast }$, here ${\mathcal{E}}_l$ corresponds to the last of CCPEs. If $V(x)$ has Kato’s exceptional points (EPs: $V_{EP1}<V_{EP2}<V_{EP3}<\mathrm{.}\mathrm{.}\mathrm{.}<V_{EPl}$), the smallest of critical values $V_{\ast m}$ is always larger than $V_{EPl}$. Hence, in a CPTSSP, real discrete eigenvalue(s) and the SS are mutually exclusive whereas CCPEs and the SS can co-exist .