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Journal ArticleDOI

Application of supersymmetry to a coupled system of equations: the concept of a superpotential matrix

26 Mar 1999-Journal of Physics A (IOP Publishing)-Vol. 32, Iss: 12, pp 2387-2394
TL;DR: In this article, supersymmetric quantum mechanics (SUSY QM) is applied to multidimensional Schrodinger equations involving nonseparable potentials, which result in a system of coupled differential equations.
Abstract: We apply supersymmetric quantum mechanics (SUSY QM) to multidimensional Schrodinger equations involving nonseparable potentials, which result in a system of coupled differential equations, where instead of the conventional definition of a scalar superpotential we introduce a superpotential matrix and succeed in applying SUSY QM to the coupled system. Finally, we discuss the shape-invariance condition for the potential matrix of such a system.
Citations
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Journal ArticleDOI
TL;DR: In this paper, a nonlinear (polynomial, N-fold) SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed, and the full classification of ladder-reducible and irreducibly chains of sUSY algebras in one-dimensional QM is given.
Abstract: A nonlinear (polynomial, N-fold) SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Possible extensions of SUSY in one dimension are described. They include (no more than) N = 2 extended SUSY with two nilpotent SUSY charges which generate the hidden symmetry acting as a central charge. Embedding stationary quantum systems into a non-stationary SUSY QM is shown to yield new insight into quantum orbits and into spectrum generating algebras.

117 citations

Journal ArticleDOI
TL;DR: The nonlinear supersymmetric (SUSY) approach to spectral problems in quantum mechanics (QM) is reviewed in this paper, its building from the chains (ladders) of linear SUSY systems is outlined and different one-dimensional and two-dimensional realizations are described.
Abstract: The nonlinear supersymmetric (SUSY) approach to spectral problems in quantum mechanics (QM) is reviewed. Its building from the chains (ladders) of linear SUSY systems is outlined and different one-dimensional and two-dimensional realizations are described. It is elaborated how the nonlinear SUSY approach provides two new methods of SUSY separation of variables for various two-dimensional models. In the framework of these methods, a partial and/or complete solution of some two-dimensional models becomes possible. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. The emergence of hidden symmetries and spectrum generating algebras is elucidated in the context of the nonlinear SUSY in one-dimensional stationary and non-stationary, as well as in two-dimensional QM.

100 citations

Journal ArticleDOI
TL;DR: In this article, the spectral properties of SUSY algebras in one-dimensional and two-dimensional quantum systems with pre-planned spectral properties are reviewed. And the full classification of ladder-reducible and irreducibly chains of SUsY algesbras is given.
Abstract: Nonlinear SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. Possible multidimensional extensions of Nonlinear SUSY are described. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Emergence of hidden symmetries and spectrum generating algebras is elucidated in the context of Nonlinear SUSY in one- and two-dimensional QM.

71 citations

Journal ArticleDOI
TL;DR: In this paper, a class of shape-invariant bound-state problems which represent two-level systems is introduced, and the coupled-channel Hamiltonians obtained correspond to the generalization of the Jaynes-Cummings Hamiltonian.
Abstract: A class of shape-invariant bound-state problems which represent two-level systems are introduced. It is shown that the coupled-channel Hamiltonians obtained correspond to the generalization of the Jaynes-Cummings Hamiltonian.

41 citations

Journal ArticleDOI
TL;DR: In this article, the superpotential and the two-component wave functions of the ground state of a two-dimensional physical system were found out using super-ymmetry in non-relativistic quantum mechanics.

21 citations

References
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Journal ArticleDOI
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.

2,688 citations

Journal ArticleDOI
TL;DR: The first-order differential-difference factorization method as mentioned in this paper is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists.
Abstract: The factorization method is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists. The underlying idea is to consider a pair of first-order differential-difference equations which are equivalent to a given second-order differential equation with boundary conditions. For a large class of such differential equations the method enables us to find immediately the eigenvalues and a manufacturing process for the normalized eigenfunctions. These results are obtained merely by consulting a table of the six possible factorization types.The manufacturing process is also used for the calculation of transition probabilities.The method is generalized so that it will handle perturbation problems.

1,536 citations

Journal ArticleDOI
TL;DR: In this paper, it is shown that the harmonic oscillator potential can be solved by using raising and lowering operators, which can be generalized with the help of supersymmetry and the concept of shape invariant potentials, allowing one to calculate energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.
Abstract: It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of ‘‘shape‐invariant’’ potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.

379 citations

Journal ArticleDOI
TL;DR: In this paper, a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions was investigated.
Abstract: The author investigates a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions. It turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions, satisfying linear homogeneous second-order differential equations, can be solutions of the Schrodinger equation with potentials of the form V(x)=W2(x)-W'(x). The author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape-invariant potentials.

258 citations