The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential
iV,
V∈R+. We show that these loci are elliptical and self-similar to each other in
E−V plane where
E is energy of the wave. The orientations of these ellipses for reflection (
R) and transmission (
T) amplitude around a given SS are same. This is surprising due to the different mathematical forms of
R and
T. In the vicinity of SS,
R≈T.
A curious feature of complex scattering potentials v ( x ) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., , and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v ( x ) = z − δ( x + a ) + z + δ( x − a ), where z ± and a are respectively complex and real parameters and δ( x ) is the Dirac delta function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing -symmetry.
A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., − d2 dx 2 +v(x), and its adjoint. We establish the equivalence of this description with the mathematicians ’ definition of spectral singularities for the potential v(x) = z−δ(x + a) + z+δ(x − a), where z ± and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry. PACS numbers: 03.65.-w
For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R ( − k ) ≠ R ( k ) and T ( − k ) ≠ 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 1. We present a new non-Hermitian parametrization of the 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 * = α 2 , β 2 ) either in T ( k ) or in T ( − k ), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T ( k ) and the other in T ( − k ). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we obtain T ( k ) = T ( − k ), it possesses a unique SS at E = α 2 in both T ( − k ) and T ( k ). Lastly, for completeness, when α = iγ and β = iδ there are no SS, instead we get two real energies −γ 2 and −δ 2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound-state eigenvalues. We find them as E + M = −(γ − M ) 2 and E − N = −(δ − N ) 2 ; M ( N ) = 0, 1, 2, ... with 0 ⩽ M ( N ) < γ(δ).
For complex
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
V2=V∗. Based on four analytically solvable and other numerically solved models, we conjecture that a parametrically fixed CPTSSP has at most one SS. When
V1 is fixed and
V2 is varied there may exist Kato's exceptional point(s)
(VEP) and critical values ${V}_{*m},\phantom{\rule{0.28em}{0ex}}m=0,1,2,\ensuremath{\cdots}$, so when
V2 crosses one of these special values a new CCPE is created. When
V2 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:
El⪅E∗, here
El 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
VEPl. Hence, in a CPTSSP, real discrete eigenvalue(s) and the SS are mutually exclusive whereas CCPEs and the SS can coexist.
When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection
(rL,rR) and transmission
(tL,tR) amplitudes are nonreciprocal. In a parametric domain, there exists spectral singularity (SS) at a real energy
E=E∗=k∗2 and the determinant of the time-reversed two port scattering matrix, i.e., $|det(S(\ensuremath{-}k))|=|{t}_{L}(\ensuremath{-}k){t}_{R}(\ensuremath{-}k)\ensuremath{-}{r}_{L}(\ensuremath{-}k){r}_{R}(\ensuremath{-}k)|$, vanishes sharply at
k=k∗, displaying the phenomenon of coherent perfect absorption (CPA). In the complementary parametric domain, the potential becomes either left or right reflectionless at
E=Ez. We rule out the existence of invisibility despite
rR(Ei)=0 and
tR(Ei)=1 but $T({E}_{i})\ensuremath{\ne}1$, in this avenue. We present two simple exactly solvable models where expressions for
E∗,
Ez,
Ei, and parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and have
tL=tR.