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| Name: | ||||||||||||||||||||
| Miloslav Znojil | ||||||||||||||||||||
| Reviewer number: | ||||||||||||||||||||
| 9689 | ||||||||||||||||||||
| Email: | ||||||||||||||||||||
| znojil@ujf.cas.cz | ||||||||||||||||||||
| Item's zbl-Number: | ||||||||||||||||||||
| DE 017 400 975 | ||||||||||||||||||||
| Author(s): | ||||||||||||||||||||
| Bagchi, B.; Mallik, S.; Quesne, C.: | ||||||||||||||||||||
| Shorttitle: | ||||||||||||||||||||
| Complexified PSUSY and SSUSY interpretations of some PT-symmetric Hamiltonians. | ||||||||||||||||||||
| Source: | ||||||||||||||||||||
| Int. J. Mod. Phys. A 17, No. 1, 51 - 72 (2002). | ||||||||||||||||||||
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Keywords:
| 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 | Review: | 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]. \par 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 origin). 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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