One of the authors of this - extremely interesting - paper [P1]
produces his results so quickly that the citation of the completed
subsequent paper [P2] is already available as ref. [19] in [P1]
(also note that another progress report [P3] is announced at the end
of the text). This speed is, unfortunately, felt. I.a., it forces me
to make this extended abstract a bit more explicatory
(alternatively, interested readers might skip my forthcoming
explanations and have a look at the more matured next-step progress
report [P2] instead).
Firstly: paper [P1] is about the structure of the spectrum of the 1D
Hamiltonian with the complex double-well delta-function potential
given in Eq. (3). In this sense, even the title would deserve an
amendment (its dominant first line carries just a complementary
information). Personally I would also like to attract attention to
my own paper on the same model (M. Znojil, J. Phys. A: Math. Gen. 36
(2003) 7639-48) and to its early practical use in connection with
certain strong-coupling expansions. Nowadays, the area of
applications of similar models is much broader of course, involving
even experiments in nonlinear optics (so that references [3,4] would
deserve an update).
Secondly, also the abstract admits an ambiguous reading. Certainly,
the paper's main message is not about a 1D quantum problem of
scattering but rather about its non-unitary (though still very
interesting) mathematical generalization. This class of methodically
relevant models (which cannot directly be assigned the standard
quantum-mechanical meaning in general) has been called ``effective",
e.g., by the author of the second item in ref. [13] (say, in his
other, much more relevant paper with coordinate D 77 065023 merely
replaced by D 78 065032).
All this being said let me emphasize that although the whole text
(having, by the way, its half-line-model immediate predecessor in
Ref. [20]) makes an impression of a hastily finished draft and
although it is difficult to read it in places, it certainly deserves
to be read. The main reason is that it offers an almost exhaustive
account of the key features of the spectrum of the four-parametric
Hamiltonian of Eq. (3). They are illustrated by 11 figures showing,
predominantly, the position of spectral singularities (= real zeros
of Jost functions) and of ``bound states" (= certain complex zeros
of Jost functions), the knowledge of which can lead immediately to
the specification of the boundaries of (quasi-)Hermiticity of the
model.
MR2485827 Mostafazadeh, Ali; Mehri-Dehnavi, Hossein Spectral
singularities, biorthonormal systems and a two-parameter family of
complex point interactions. J. Phys. A 42 (2009), no. 12, 125303, 27
pp. 81Uxx
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1. Similar potentials proved much less interesting in the context of
the standard quantum theory of scattering where their applications
seem to encounter enormous technical obstacles -- this has been
shown by H. F. Jones, one of the ``fathers founders" of PT symmetric
quantum mechanics, in his papers in Phys. Rev. D 76, 125003 (2007)
and in Phys. Rev.D 78, 065032 (2008).
2. Marginally, one should add that the latter version of quantum
theory has also been known, especially in the context of nuclear
physics, as quasi-Hermitian quantum mechanics of bound states (cf.,
e.g., paper [8] or my own recent compact review arXiv:0901.0700).
3. Marginally, I would like to add that the first complexification
of delta-function potentials appeared, in [27], as one of
illustrations of the innovative formalism of the so called PT
symmetric quantum mechanics of bound states [cf. the pioneering
letter C. M. Bender D. C. Brody and H. F. Jones, Phys. Rev. Lett.
89, 270402 (2002)].
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