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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 400 975
Bagchi, B.; Mallik, S.; Quesne, C.:
Complexified PSUSY and SSUSY interpretations of some PT-symmetric Hamiltonians.
Int. J. Mod. Phys. A 17, No. 1, 51 - 72 (2002).
34M15Algebraic aspects differential-algebraic, hypertranscendence, group-theoretical
Primary Classification:
81Q60Supersymmetric quantum mechanics
Secondary Classification:
81Q05Closed and approximate solutions to the Schroedinger, Dirac, Klein-Gordon and other quantum-mechanical equations
order-two parasupersymmetry; second-derivative supersymmetry; complexified potentials of Scarf, Poeschl-Teller and spiked harmonic oscillator; PT symmetric models in regime with real spectra; pseudo-Hermiticity; two subspectra of a given quasi-parity
The setting of the scene:
The elementary and well known lowering of the order of the linear
differential Schroedinger equation in one dimension (dating back
to the names of Riccati and Darboux) found a modern re-wording
within the Witten's supersymmetric (= SUSY) quantum mechanics. Due
to the changes of emphasis, this type of symmetry succeeded in
elucidating some aspects of the relationship between the quantized
bosonic and fermionic fields and
it offered new keys to the problem of classification of the
exactly solvable potentials. At the same time it failed to work
for the Riccatian ground-state solutions W(x) (Witten calls them
superpotentials) with singularities. It has been noticed in
1999 that the transition to the Bender's (often called PT
symmetric or pseudo-Hermitian) quantum mechanics
regularizes the singularities in W(x) and replaces the common parity
quantum number by the so called quasi-parity (cf. ref. [9]). This
broadens the class of the admissible superpotentials W(x) and assigns
the two different partner potentials to a given shape-invariant
initial interaction in a way which was noticed in the year 2000
[cf. the Los Alamos preprint arXiv: hep-th/0012002 published in
J. Phys. A: Math. Gen. 35 (2002) 2341].
A very compact and comprehensive review of the formalism of
the standard, Hermitian SUSY quantum mechanics has been
written in 1995 by Cooper, Khare and Sukhatme (Phys. Reports, vol.
251, p. 267). In this context,
the present paper offers the excellent next-step reading which
reflects the new progress achieved after
the extension of the scope of quantum mechanics by Bender et al in
1998 (cf. ref. [2]).
In particular, the above-mentioned ''double-partner" phenomenon
(intensively studied, in parallel, in arXiv: quant-ph/0206013 etc) is
being given its new, deep and nice
alternative algebraic explanation within the formalisms of the so
called para-SUSY (PSUSY) and of the second-order SUSY (SSUSY).
In the former case the two alternative partner Hamiltonians
correspond to the so called para-fermionic field [20]. Their
subsequent SSUSY re-arrangement (cf. [23]) explains the
same effect within different approach. The presented paper gives
the details of these two algebraic re-interpretations of solutions for the
PT symmetric form of the spiked harmonic oscillator and for the
two asymptotically vanishing PT symmetric
potentials of Poeschl-Teller (in its so called generalized version)
and of Scarf (hyperbolic form II which is regular in the
Although the authors mainly emphasize the interesting immediate
correspondence between the symmetries and the spectra, one could
also appreciate another, ''hidden" merit of their
two constructions which show that and how the symmetry
pattern can survive a transition to its much less standard
representation in terms of non-Hermitian operators.
Remarks to the editors:

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