For a free quantum particle living on an interval of $x \in (-1,1)$
the Sturm-Liouville oscillation theorems tell us that the wave
functions and their derivatives can only match in a specific manner
in the origin. The authors make use of this fact and introduce a
trivial spectrum-preserving two-parametric asymmetry in the model.
This produces a Hamiltonian $H$ which is/looks manifestly
non-self-adjoint with respect to the ``usual" inner product [i.e.,
in the ``false", unphysical Hilbert space ${\cal H}^{(F)}$ - I am
using and recommending here my own notation as well as detailed
comments available in MZ, SIGMA 5 (2009), 001
(doi:10.3842/SIGMA.2009.001)]. Then the authors derive a formula for
the metric operator in the correct physical Hilbert space {$={\cal
H}^{(S)}$ in my notation).

Mathematics Subject Classifications (2000): 34L05, 81Q05, 8IQlO.

Keywords: non-Hermitian, pseudo-Hermitian, metric operator, complete
system.

MR2677509 Ergun, Ebru; Saglam, Mesude On the metric of a
non-Hermitian model. Rep. Math. Phys. 65 (2010), no. 3, 367--378.
81Q80 (34L05 81Q12)