In the general framework of Quantum Mechanics of exactly solvable
models (keeping the coordinates, for the sake of definiteness,
real) our contemporary improvement of understanding of {\em
complex} potentials leading to {\em real} (i.e., in principle,
observable) energies proceeds in three steps. In the prehistory of
the subject the first-step discoveries have been made of a close
relationship between the reality of the spectrum and a Wick's
rotation of the antisymmetric part of the potential. The
second-step results are characteristic for the present stage of
development. They are well exemplified in ref. [30] or by the
L\'evai's paper in question. People (cf. the list of references)
calculate the pseudonorms of bound states in closed form. In
particular, L\'{e}vai picks up the model with the name Rosen-Morse
I and, in detail, he discusses some of the most important
properties of the pseudonorms, e.g., their oscillatory
excitation-dependence or their change after the backward Wick's
rotation. All thiese efforts prepare the terrain for the final
construction of the metric operator $\Theta$. In this third-step
activity the knowledge of the pseudonorms will prove vital,
opening the way towards a climax of the story in the nearest
future. Indeed, operator $\Theta$ will characterize physics and
measurements. Equivalently, in the language of mathematics one can
say that the specification of $\Theta$ will determine the
Hamiltonian-assigned, non-standard operation of Hermitian
conjugation in the correct though mathematically nontrivial
physical Hilbert space of states.
MR2464272 Lévai, G. On the normalization constant of
$\scr{PT}$-symmetric and real Rosen-Morse I potentials. Phys.
Lett. A 372 (2008), no. 43, 6484--6489. 81Qxx