Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasi-exactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified nonpolynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.
With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x1+ip2, p=p1+ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT-symmetric condition is satisfied. Thus PT- symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.
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.