We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the author missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Bender 2007 Rep. Prog. Phys. 70 947)
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.
The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian
H=p2+x2(ix)ε has real, positive, and discrete eigenvalues for all
ε≥0. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues
En=2n+1 (n=0, 1, 2, 3, ...) at
ε=0. However, the harmonic oscillator also has negative eigenvalues
En=−2n−1 (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian
H=p2+x2(ix)ε also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues,
ε lies in the range $(4N-6)/3
In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.