Three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials (1806.06578v4)

Abstract: 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 $, we show that complex $k$-poles of transmission amplitude $t(k)$ or zeros of $1/t(k)$ of the type $\pm k_1+ik_2, k_2\ge 0$ 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: ${\cal E}n \pm i \gamma_n$) and real positive energy called spectral singularity (SS) at $E=E$ where the transmission and reflection co-efficient of $V(x)$ become infinite for a special critical value of $V_2=V_$. 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 $V_{m}, m=0,1,2,..$, so when $V_2$ crosses one of these special values a new CCPE is created. When $V_2$ equals a critical value $V_{*m}$ there exist one SS at $E=E_$ along with $m$ or more number of CCPEs. Hence, this single positive energy $E_$ is the upper (or rough upper) bound to the CCPEs: ${\cal E}l \lessapprox E$, here ${\cal E}l$ corresponds to the last of CCPEs. If $V(x)$ has Kato's exceptional points (EPs: $V{EP1}<V_{EP2}<V_{EP3}<...<V_{EPl}$), the smallest of critical values $V_{*m}$ 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 .