Darboux transformations for the generalized Schrödinger equation

Suzko, A. A.; Giorgadze, G.

doi:10.1134/s1063778807030234

Cited by 17 publications

(26 citation statements)

References 6 publications

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“…q(x) = 1, from our supersymmetry approach we get the supersymmetry for the effective mass Schrödinger equation [23,25]. In the case with constant mass m(x) = m 0 from our approach we obtain the supersymmetry for Schrödinger equation with weighted energy [34]. Finally, if m(x) = m 0 and q(x) = 1, our expressions of supersymmetric algebra are cor rectly reduced to the conventional ones for the stan dard Schrödinger equation (see, e.g.…”

Section: Supersymmetrymentioning

confidence: 97%

“…It has been constructed for a variety of linear and nonlin ear equations. Among the linear equations to which the Darboux transformations are applicable, there are Pauli and Dirac equations [4,7,8], conventional Schrödinger equations, see the reviews in [8][9][10][11] the class of Schrödinger equations with position depen dent effective mass [22][23][24][25][26] and Schrödinger type equations with energy dependent potentials [15,34]. The Darboux transformations is essentially based on intertwining relations and differ from other transfor 1 The article is published in the original.…”

Section: Introductionmentioning

confidence: 98%

“…The efforts of many authors have been concentrated on the inverse eigenvalue problems on the determination of q(x) and v(x) from the eigen values of the operator [43][44][45][46][47][48], One can cite more recent investigations, related to a polynomial form of potential coefficients [49,52,53] in the second order differential operator and as well as the intertwining technique for wave equations with linearly energy dependent potentials [34].…”

Section: Introductionmentioning

confidence: 99%

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Intertwining relations and Darboux transformations for the wave equations

Suzko

Velicheva

2012

Phys. Part. Nuclei

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Using the intertwining operator technique we construct Darboux transformations for the wave equation with position dependent effective mass and with linearly energy dependent potentials. The for mally adjoint generators of supersymmetry and two formally self adjoint superpartner Hamiltonians are con structed and they close a quadratic pseudo superalgebra. The Darboux transformations are constructed in differential and integral forms and an interrelation is established between them. The approach is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials without changing the spectrum, i.e. fully isospectral potentials. The method is illustrated by some concrete examples. The influence of distance between levels on the form of potentials is investigated. In par ticular, asymmetric double well potentials are generated.

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Section: Supersymmetrymentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Intertwining relations and Darboux transformations for the wave equations

Suzko

Velicheva

2012

Phys. Part. Nuclei

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“…In the stationary case, where the time-derivative is substituted by a multiplication with the energy, the nonconstant factor can be seen as weighting the energy [16], or as generating an energy-dependent potential [17]. For the sake of generality, we consider here the fully time-dependent case.…”

Section: Tdse With Weighted Energymentioning

confidence: 99%

Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions

Schulze-Halberg

Jimenez

2009

Symmetry

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We review recent results on how to extend the supersymmetry SUSY formalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, such as the Schrödinger equation for position-dependent mass. By evaluating intertwining relations, we obtain explicit formulas for the interrelations between the supersymmetric partner potentials and their corresponding solutions. We review reality conditions for the partner potentials and show how our SUSY formalism can be extended to the Fokker-Planck and the nonhomogeneous Burgers equation.

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“…the Dirac equation [3,4], the nonlinear Schrödinger equation [5] (in fact here all classical nonlinear equations are considered), the Korteweg-de-Vries equation [6], the sine-Gordon equation [6] and many more. Among these equations that allow for Darboux transformations there is the class of linear generalizations of the Schrödinger equation, such as Schrödinger equations with positiondependent (effective) mass [7], Schrödinger equations coupled to a magnetic field [8] and Schrödinger equations with weighted energy [9]. All these equations are linear and of second order, and all of them allow for the construction of certain Darboux transformations.…”

Section: Introductionmentioning

confidence: 99%