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arXiv:1806.06578 · Physical Review A

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

Zafar Ahmed, Sachin Kumar, D. C. Ghosh

Published Oct 1, 2018• 7 citations
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Abstract

For complex $\mathit{PT}$-symmetric scattering potentials (CPTSSPs) $V(x)={V}_{1}{f}_{\mathrm{even}}(x)+i{V}_{2}{f}_{\mathrm{odd}}(x),\phantom{\rule{0.28em}{0ex}}{f}_{\mathrm{even}}(\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty})=0={f}_{\mathrm{odd}}(\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}),\phantom{\rule{0.28em}{0ex}}{V}_{1},\phantom{\rule{0.28em}{0ex}}{V}_{2}\ensuremath{\in}\mathrm{Re}$, we show that complex $k$ poles of transmission amplitude $t(k)$ or zeros of $1/t(k)$ of the type $\ifmmode\pm\else\textpm\fi{}{k}_{1}+i{k}_{2},\phantom{\rule{0.28em}{0ex}}{k}_{2}\ensuremath{\ge}0$ 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: ${\mathcal{E}}_{n}\ifmmode\pm\else\textpm\fi{}i{\ensuremath{\gamma}}_{n}$) and real positive energy called spectral singularity (SS) at $E={E}_{*}$ where the transmission and reflection coefficient 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) $({V}_{\mathrm{EP}})$ and critical values ${V}_{*m},\phantom{\rule{0.28em}{0ex}}m=0,1,2,\ensuremath{\cdots}$, 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 exists 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: ${\mathcal{E}}_{l}⪅{E}_{*}$, here ${\mathcal{E}}_{l}$ corresponds to the last of CCPEs. If $V(x)$ has Kato's exceptional points (EPs: ${V}_{\mathrm{EP}1}<{V}_{\mathrm{EP}2}<{V}_{\mathrm{EP}3}<\ensuremath{\cdots}<{V}_{\mathrm{EPl}}$), the smallest of critical values ${V}_{*m}$ is always larger than ${V}_{\mathrm{EPl}}$. Hence, in a CPTSSP, real discrete eigenvalue(s) and the SS are mutually exclusive whereas CCPEs and the SS can coexist.

DOI
10.1103/physreva.98.042101

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