# Generating isospectral hamiltonians from a modified crum–darboux transformation

20 Aug 1998-International Journal of Modern Physics A (World Scientific Publishing Company)-Vol. 13, Iss: 21, pp 3711-3721

TL;DR: Using supersymmetry as a tool, this paper set up a procedure to generate a sequence of pairwise isospectral potentials from a modified Crum-Darboux transformation, which gives rise to shifted superpotentials much similar to what we encounter in a modified factorization scheme.

Abstract: Using supersymmetry as a tool we set up a procedure to generate a sequence of pairwise isospectral potentials from a modified Crum–Darboux transformation. We show that such a transformation gives rise to shifted superpotentials much similar to what we encounter in a modified factorization scheme. We also explore some solvable systems to illustrate the applicability of our scheme.

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TL;DR: In this paper, the authors construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrodinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials.

Abstract: We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrodinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.

286 citations

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TL;DR: In this article, the authors construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrodinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials.

Abstract: We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schr\"odinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.

200 citations

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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.

95 citations

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TL;DR: In this paper, a second-order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass, and the intertwining relations between secondorder partner Hamiltonians may be exploited to obtain a simple shape-invariant condition.

Abstract: A second-order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second-order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed, a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable models. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscillator-like or singular-oscillator-like spectra depending on the values of the shape-invariant parameter.

68 citations

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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.

65 citations

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TL;DR: In this article, general conditions for dynamical supersymmetry breaking are discussed and examples are given (in 0 + 1 and 2 + 1 dimensions) in which such a program in four dimensions is possible.

Abstract: General conditions for dynamical supersymmetry breaking are discussed. Very small effects that would usually be ignored, such as instantons of a grand unified theory, might break supersymmetry at a low energy scale. Examples are given (in 0 + 1 and 2 + 1 dimensions) in which dynamical supersymmetry breaking occurs. Difficulties that confront such a program in four dimensions are described.

3,164 citations

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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.

1,476 citations

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TL;DR: Crum's seminal result of 1955 is archived here as mentioned in this paper and can be viewed as a starting point for the present paper, and is discussed in Section 5.2.1.

Abstract: Crum's seminal result of 1955 is archived here

562 citations

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TL;DR: In this article, a one-parameter family of potentials in one dimension was constructed with the energy spectrum coinciding with that of the harmonic oscillator, which is a new derivation of a class of possible potentials previously obtained by Abraham and Moses with the help of the Gelfand-Levitan formalism.

Abstract: A one‐parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator. This is a new derivation of a class of potentials previously obtained by Abraham and Moses with the help of the Gelfand–Levitan formalism.

362 citations

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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.

354 citations