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Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

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  • Kravchenko, Vladislav V.
  • Navarro, Luis J.
  • Torba, Sergii M.

Abstract

A new representation of solutions to the equation −y′′+q(x)y=ω2y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter ω. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to ω which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

Suggested Citation

  • Kravchenko, Vladislav V. & Navarro, Luis J. & Torba, Sergii M., 2017. "Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 173-192.
    • Handle: RePEc:eee:apmaco:v:314:y:2017:i:c:p:173-192
    DOI: 10.1016/j.amc.2017.07.006
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    References listed on IDEAS

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    1. Kravchenko, Vladislav V. & Morelos, Samy & Torba, Sergii M., 2016. "Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 321-336.
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    Cited by:

    1. Khmelnytskaya, Kira V. & Kravchenko, Vladislav V. & Torba, Sergii M., 2019. "A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 274-281.
    2. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & Jos'e Carlos Dias, 2017. "Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation," Papers 1712.08247, arXiv.org.
    3. Kravchenko, Vladislav V., 2018. "Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 75-81.
    4. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & José Carlos Dias, 2019. "Pricing Double Barrier Options On Homogeneous Diffusions: A Neumann Series Of Bessel Functions Representation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-24, September.
    5. Gutiérrez Jiménez, Nelson & Torba, Sergii M., 2020. "Spectral parameter power series representation for solutions of linear system of two first order differential equations," Applied Mathematics and Computation, Elsevier, vol. 370(C).

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    1. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & Jos'e Carlos Dias, 2017. "Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation," Papers 1712.08247, arXiv.org.
    2. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & José Carlos Dias, 2019. "Pricing Double Barrier Options On Homogeneous Diffusions: A Neumann Series Of Bessel Functions Representation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-24, September.

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