New features of scattering from a one-dimensional non-Hermitian (complex) potential

Abstract

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: $R(-k)\ne R(k)$ and $T(-k)\ne T(k)$, unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that $R_{left}(-k)=R_{right}(k)$ and $T(-k)=T(k)$. So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ($E_{\ast }={\alpha }^2,{\beta }^2$) either in $T(k)$ or in $T(-k)$, when $\alpha \beta >0$. Thirdly, when $\alpha \beta <0$ it possesses one SS in $T(k)$ and the other in $T(-k)$. Fourthly, when the potential becomes PT-symmetric $[(\alpha +\beta )=0]$, we get $T(k)=T(-k)$, it possesses a unique SS at $E={\alpha }^2$ in both $T(-k)$ and $T(k)$. Lastly, for completeness, when $\alpha =i\gamma$ and $\beta =i\delta$, there are no SS, instead we get two negative energies $-{\gamma }^2$ and $-{\delta }^2$ of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound state eigenvalues and no spectral singularity exists in this case. We find them as $E_M^{+}=-(\gamma -M{)}^2$ and $E_N^{-}=-(\delta -N{)}^2$; $M(N)=0,1,2,...$ with $0\le M(N)<\gamma (\delta )$. {PACS: 03.65.Nk,11.30.Er,42.25.Bs}

10 pages, one Table, one Figure, important changes, appeared as an FTC (J. Phys. A: Math. Theor. 45(2012) 032004)

References in corpus (2)

• Making Sense of Non-Hermitian Hamiltonians
• Visualization of Branch Points in PT-Symmetric Waveguides

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