Exactly solvable combined potentials and their isospectral partners from SUSY

Authors

  • J. J. Peña,

    Corresponding author
    1. Division de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Azcapotzalco, Av. San Pablo 180, 02200 México, D. F.
    • Division de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Azcapotzalco, Av. San Pablo 180, 02200 México, D. F.
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  • M. A. Romero-Romo,

    1. Division de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Azcapotzalco, Av. San Pablo 180, 02200 México, D. F.
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  • J. Morales,

    1. Division de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Azcapotzalco, Av. San Pablo 180, 02200 México, D. F.
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  • J. L. López-Bonilla

    1. Escuela Superior de Ingenería Mecánica y Eléctrica, Estudios de Posgrado e Investigación, Edificio Z-4 3er Piso, Instituto Politécnico Nacional, Zacatenco, Unidad Profesional Adolfo López Mateos, 07738 México, D. F.
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Abstract

In some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case, for example, of the equivalent of the Stark effect for a charged harmonic oscillator (HO) in a constant electric field or an HO in an external dipole field. The same occurs with some molecular systems where the determination of the vibrational spectra remains essentially as a numerical task; the Morse and Hulthén potentials are convenient models for the potential energy of a diatomic molecule. In this work, the supersymmetry (SUSY) and shape invariance methods used to determine exactly solvable potentials are extended to obtain, in a closed form, the eigenvalues of exactly solvable combined potentials, as well as their isospectral or generalized partner potentials. Essentially, the proposal is based on the existence of the Witten superpotential [W(x)] associated with single exactly solvable potentials to find the Witten superpotential for the combined potential under study. For instance, the assumption of two W(x) allows to obtain an exactly solvable combined potential which is given by a linear combination between the former potentials plus a term resulting from the product of the two single W(x). The corresponding generalized partner potentials comes from the involved Riccati equation when the resulting Witten superpotential for the combined potential is taken as a specific solution. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

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