Two-step shape invariance in the framework of $$\mathcal{N}$$ -fold supersymmetry

Roy, Barnana; Tanaka, Toshiaki

doi:10.1140/epjp/i2015-15025-5

Cited by 3 publications

(8 citation statements)

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“…Regarding the subject of intermediate Hamiltonians in N -fold SUSY, we would like to recall its relevance on the concept of shape invariance [16] and its muti-step generalization [17], which have been practical methods in constructing solvable quantum Hamiltonians. Not only was ordinary shape invariance treated efficiently in the type A 2-and 3-fold SUSY with intermediate Hamiltonians [11,15], but two-step one was also classified systematically in the framework of type A 2-fold SUSY [18]. The next scope in this field is definitely three-step generalization, and our prospect is that the type B 3-fold SUSY formalism with the present GL(3, C) equipment would allow us to develop farsightedly the issue.…”

Section: Discussion and Summarymentioning

confidence: 99%

$GL(3,\mathbb {C})$ G L ( 3 , C ) invariance of type B 3-fold supersymmetric systems

Tanaka

2014

Journal of Mathematical Physics

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Type B 3-fold supersymmetry is a necessary and sufficient condition for a quantum Hamiltonian to admit three linearly independent local solutions in closed form. We show that any such a system is invariant under G L(3, C) homogeneous linear transformations. In particular, we prove explicitly that the parameter space is transformed as an adjoint representation of it and that every coefficient of the characteristic polynomial appeared in 3-fold superalgebra is algebraic invariants. In the type A case, it includes as a subgroup the G L(2, C) projective transformation studied in the literature. We argue that any N -fold supersymmetric system has a G L(N , C) invariance for an arbitrary integral N . C 2014 AIP Publishing LLC.

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Section: Discussion and Summarymentioning

confidence: 99%

$GL(3,\mathbb {C})$ G L ( 3 , C ) invariance of type B 3-fold supersymmetric systems

Tanaka

2014

Journal of Mathematical Physics

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“…In the case where the number is two, it was proved [12] that a necessary and sufficient condition for it is to have type A 2-fold SUSY. The latter fact further enabled one to construct systematically shape-invariant potentials within this type of symmetry [13,14]. These achievements clearly show the power of N -fold SUSY, and we expect that we would be able to step forward to more general cases, beginning with the case where the number of available analytic solutions is three.…”

Section: Introductionmentioning

confidence: 91%

“…[24], it was generalized to the notion of two-and multi-step shape invariance, keeping the sufficiency for solvability intact. Recently, it was shown [14] that any two-step shape-invariant model can be systematically investigated and constructed in the framework of N -fold SUSY as a particular case of 2-fold SUSY with an intermediate Hamiltonian developed in Ref. [13].…”

Section: Type X 2 3-fold Susy With Polynomial Subspacesmentioning

confidence: 99%

Type B 3-fold supersymmetry and non-polynomial invariant subspaces

Tanaka

2013

Journal of Mathematical Physics

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We obtain the most general type B 3-fold supersymmetry by solving directly the intertwining relation. We then show that it is a necessary and sufficient condition for a second-order linear differential operator to have three linearly independent local analytic solutions. We find that there are eight linearly independent non-trivial linear differential operators of this kind. As a by-product, we find new quasi-solvable second-order operators preserving a monomial or polynomial subspace, one in type B, two in type C, and four in type X 2 , all of which have been missed in the existing literature. In addition, we show that type A, type B, and type C 3-fold supersymmetries are connected continuously via one parameter. A few new quasi-solvable models are also presented. C 2013 AIP Publishing LLC. [http://dx.

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“…the discrete spectrum of the model. The wave functions (30), specified by (31) and reference mode (40) must be substituted into (42):…”

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confidence: 99%

“…In contrast to this approach, in the present paper in terms of W we considered actually the "square root dependence" of W via √ −d ≡ √ a (see (15)). As for the [30], many potentials with two-step…”

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New implicitly solvable potential produced by second order shape invariance

et al. 2015

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The procedure proposed recently by J.Bougie, A.Gangopadhyaya and J.V.Mallow [1] to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.Recently, the general investigation of additive form of shape invariance for superpotentials without explicit dependence on parameters was performed by J.Bougie et al [1], [11]. It was demonstrated that no additional shape invariant models in one-dimensional case can be constructed. This result was obtained in the framework of standard SUSY QM with supercharges of first order in derivatives. Meanwhile, it is known [18], [19], [20] that such standard SUSY QM does not exhaust all opportunities to fulfill the generalized SUSY algebra for SuperhamiltonianĤ and superchargesQ ± . Higher order supercharges are also possible, and for example supercharges of second order in momentum lead to some new results [20].

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Two-step shape invariance in the framework of $$\mathcal{N}$$ -fold supersymmetry

Cited by 3 publications

References 47 publications

$GL(3,\mathbb {C})$ G L ( 3 , C ) invariance of type B 3-fold supersymmetric systems

$GL(3,\mathbb {C})$ G L ( 3 , C ) invariance of type B 3-fold supersymmetric systems

Type B 3-fold supersymmetry and non-polynomial invariant subspaces

New implicitly solvable potential produced by second order shape invariance

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