Via additional examples, the authors illustrate the practical
applicability of the method proposed in their previous paper [1] (=
recommended reading). Their very inspiring idea is based on the
assumption of our knowledge of the so called generator $\sigma_E$ of
the eigenstates of a given Hamiltonian $H$.

The present examples include the trivial 2 by 2 toy model and the
nontrivial imaginary cubic perturbation of harmonic oscillator. In
the latter context it is extremely unfortunate that the authors did
not notice the existence of the highly relevant, very strongly
recommended and probably entirely general result published in paper
``Metric Operator in Pseudo-Hermitian Quantum Mechanics and the
Imaginary Cubic Potential" by Ali Mostafazadeh in J. Phys. A: Math.
Gen. 39 (2006) 10171-10188 (arXiv:quant-ph/0508195).

MR2662467 Das, Ashok; Greenwood, L. An alternative construction of
the positive inner product for pseudo-Hermitian Hamiltonians:
examples. J. Math. Phys. 51 (2010), no. 4, 042103, 11 pp. 81Q12