An extremely interesting new and well-developed application of the
old idea due to Scholtz et al (cited as [28]) who, in 1992,
emphasized that the internal consistency of Quantum Mechanics
remains preserved when one replaces the most common (i.e.,
Hermitian) representation of the observable quantities by its less
common (viz., quasi-Hermitian) alternative using the inner
products of the form $\langle \psi |\Theta|\psi'\rangle$ based on
a non-Dirac ``metric" $\Theta=\Theta^\dagger\neq I$ in the
``physical" Hilbert space of states. Ali Mostafazadeh picks up one
of the simplest possible assumptions about dynamics (considering
just a one-dimensional Schr\"{o}dinger equation with a single
delta function interaction) for which $\Theta \neq I$ (i.e., for
which the coupling constant becomes a complex number) and
elaborates a number of mathematical as well as physical
consequences (with emphasis on the quantitative analysis of the
smearing effects due to the imaginary part of the coupling, etc).
Strong mathematical points of the paper lie in an explicit
perturbative construction of the integral-kernel form of $\Theta$
(the neutral symbol we use for both $T$ of ref. [8] and/or for
$\eta_+$ in the AM's notation) and in a (rarely seen and usually
difficult) demonstration of the boundedness and/or correct
Hermitian limit of this operator, in its given approximate form at
least. In the context of physics, the detailed discussion of the
non-locality features of the model is equally and particularly
impressive.


MR2269700 Mostafazadeh, Ali Delta-function potential with a
complex coupling. J. Phys. A 39 (2006), no. 43, 13495--13506.
81Q10 (81Q05)