Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q=+/-1 in PT symmetric (non-Hermitian) quantum mechanical potential models, the relation of PT symmetry and supersymmetry (SUSY) is studied. As an illustrative example the PT invariant version of the Scarf II potential is presented, and it is shown that the "bosonic" Hamiltonian has two different "fermionic" SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q=1 and q=-1. It is shown that the "fermionic" potentials cease to be PT invariant when the PT symmetry of the "bosonic" potential is spontaneously broken. A modified PT symmetry inspired SUSY construction is also discussed, in which the SUSY charge operators contain the antilinear operator T. It is shown that in this scheme the "fermionic" Hamitonians are just the complex conjugate of the original "fermionic" Hamiltonians, and thus possess the same energy eigenvalues.
Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q = ± 1 in 𝒫𝒯 symmetric (non-Hermitian) quantum mechanical potential models, the relation of 𝒫𝒯 symmetry and supersymmetry (SUSY) is studied. As an illustrative example, the 𝒫𝒯 invariant version of the Scarf II potential is presented, and it is shown that the ‘bosonic’ Hamiltonian has two different ‘fermionic’ SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q = 1 and q = −1. It is shown that the ‘fermionic’ potentials cease to be 𝒫𝒯 invariant when the 𝒫𝒯 symmetry of the ‘bosonic’ potential is spontaneously broken. A modified 𝒫𝒯 symmetry inspired SUSY construction is also discussed, in which the SUSY charge operators contain the antilinear operator 𝒯. It is shown that in this scheme the ‘fermionic’ Hamiltonians are just the complex conjugates of the original ‘fermionic’ Hamiltonians, and thus possess the same energy eigenvalues.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form
H=p2+V(x), where
V(x) is odd in
x, pure imaginary, and vanishes as
∣x∣→∞. Five PT-symmetric potentials are studied: the Scarf-II potential
V1(x)=iA1sech(x)tanh(x), which decays exponentially for large
∣x∣; the rational potentials
V2(x)=iA2x/(1+x4) and
V3(x)=iA3x/(1+∣x∣3), which decay algebraically for large
∣x∣; the step-function potential
V4(x)=iA4sgn(x)θ(2.5−∣x∣), which has compact support; the regulated Coulomb potential
V5(x)=iA5x/(1+x2), which decays slowly as
∣x∣→∞ and may be viewed as a long-range potential. The real parameters
An measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from
0 to
+∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for
Vn(x) (
1≤n≤4) the number of these pairs is finite and increases as the value of the strength parameter
An increases. However, for
V5(x) there is an {\it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.
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.
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).