Nonlinear supersymmetric quantum mechanics: concepts and realizations

Andrianov, Alexander A.; Ioffe, M. V.

doi:10.1088/1751-8113/45/50/503001

Cited by 95 publications

(160 citation statements)

References 276 publications

(913 reference statements)

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“…The polynomial two-dimensional SUSY QM allowed to build two new procedures [17], [18], [7], [5] for solution of the Schrödinger equation which can be useful when the standard separation of variables does not work: 1) Partial solvability, or quasi-exact solvability (QES) in another terminology; 2) Complete solvablity; Both methods were called as supersymmetrical separations of variables.…”

Section: Susy Qm and New Separation Of Variables: D =mentioning

confidence: 99%

“…Very soon, it became clear that SUSY QM provides the really new approach to many problems of Quantum Mechanics itself [4], [5]. Now, it is one of the most effective approaches for analytical methods in Quantum Mechanics.…”

Section: Introductionmentioning

confidence: 99%

“…The new notion of shape invariance [6] formulated in SUSY QM has to be noticed here since it allows to solve problems in a pure algebraic way. A new impetus was given to the development of multidimensional d > 1 Quantum Mechanics [5]. In particular, before this time the most known and most effective regular approach for study of d = 2 systems was the separation of variables.…”

Section: Introductionmentioning

confidence: 99%

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Shape invariance and SUSY separation of variables

2016

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Abstract. The main ingredients of conventional Supersymmetrical Quantum Mechanics (SUSY QM) are presented. The generalization with supercharges of second order in derivatives -Second Order SUSY -is formulated, and the property of shape invariance is defined. The generalization to two-dimensional coordinate space, after using just these two elements of the modern SUSY QM approach, provides the opportunity to solve analytically some two-dimensional problems. Two different procedures of supersymmetrical separation of variables are formulated. They are illustrated by two-dimensional generalization of the Morse model.

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Section: Susy Qm and New Separation Of Variables: D =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shape invariance and SUSY separation of variables

2016

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“…These equations immediately lead to the higher-order SUSY QM [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In this treatment, the standard SUSY algebra with two generators…”

Section: Higher-order Susy Qmmentioning

confidence: 99%

Supersymmetric quantum mechanics and Painlevé equations

Bermúdez

Fernández

2014

AIP Conference Proceedings

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In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order PHA the potential is determined by solutions to Painlevé IV (PIV) and Painlevé V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms of confluent hypergeometric functions. Furthermore, we will classify them into several solution hierarchies, according to the specific special functions they are connected with.

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“…Even if only for systems with a single degree of freedom, this is a field which still offers most tantalising and fascinating perspectives indeed, as witnessed by a renewed strong research activity over the last decade at least [3], which goes on unabated and keeps exploring quite many open avenues, while extensions to more degrees of freedom, including fermionic or spin ones, is a realm remaining largely unexplored until now (for a review and further references, see Ref. [4]). …”

Section: Introductionmentioning

confidence: 99%

Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials

Iyela

Govaerts

Hounkonnou

2013

Journal of Mathematical Physics

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Within the context of Supersymmetric Quantum Mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N . By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy of quantum systems which should allow for its solution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of Supersymmetric Quantum Mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper these ideas are presented and solved explicitly for the cases N = 1 and N = 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. At the same time new classes of integrable quantum potentials which generalise that of the harmonic oscillator and which are characterised by two arbitrary energy gaps are identified, for which a complete solution is achieved algebraically.

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Nonlinear supersymmetric quantum mechanics: concepts and realizations

Cited by 95 publications

References 276 publications

Shape invariance and SUSY separation of variables

Shape invariance and SUSY separation of variables

Supersymmetric quantum mechanics and Painlevé equations

Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials

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