\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the missing bound state data of inverse spectral-scattering problems on the half-line

Abstract Related Papers Cited by
  • The inverse spectral-scattering problems for the radial Schrödinger equation on the half-line are considered with a real-valued integrable potential with a finite moment. It is shown that if the potential is sufficiently smooth in a neighborhood of the origin and its derivatives are known, then it is uniquely determined on the half-line in terms of the amplitude or scattering phase of the Jost function without bound state data, that is, the bound state data is missing.
    Mathematics Subject Classification: Primary: 34A55; Secondary: 34L40, 34L20.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    P. B. Abraham and H. E. Moses, Changes in potentials due to changes in the point spectrum: anharmonic oscillators with exact solutions, Phys. Rev. A, 22 (1980), 1333-1340.doi: 10.1103/PhysRevA.22.1333.

    [2]

    T. Aktosun, Inverse Schrödinger scattering on the line with partial knowledge of the potential, SIAM J. Appl. Math., 56 (1996), 219-231.doi: 10.1137/S0036139994273995.

    [3]

    T. Aktosun and R. Weder, Inverse scattering with partial information on the potential, J. Math. Anal. Appl., 270 (2002), 247-266.doi: 10.1016/S0022-247X(02)00070-7.

    [4]

    T. Aktosun and R. Weder, Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation, Inverse Problems, 22 (2006), 89-114.doi: 10.1088/0266-5611/22/1/006.

    [5]

    I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akad. Nauk SSSR. Ser. Mat., 15 (1951), 309-360. [In Russian.]

    [6]

    F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, I. The case of an a.c. component in the spectrum, Helv. Phys. Acta, 70 (1997), 66-71.

    [7]

    F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, Trans. Amer. Math. Soc., 352 (2000), 2765-2787.doi: 10.1090/S0002-9947-99-02544-1.

    [8]

    B. Grebert and R. Weder, Reconstruction of a potential on the line that is a priori known on the half line, SIAM J. Appl. Math., 55 (1995), 242-254.doi: 10.1137/S0036139993254656.

    [9]

    M. V. Klibanov and P. E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Phys., 33 (1992), 3813-3821.doi: 10.1063/1.529990.

    [10]

    B. M. Levitan, The determination of a Sturm-Liouville operator from one or from two spectra, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), 185-199, 215-216.

    [11]

    B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1978.

    [12]

    N. N. Novikova and V. M. Markushevich, Uniqueness of the solution of the one-dimensional problem of scattering for potentials located on the positive semiaxis Vychisl, Seismol., 18 (1985), 176-184; N. N. Novikova and V. M. Markushevich, Comput. Seismol., 18 (1987), 164-172. [English Translation.]

    [13]

    V. A. Marchenko, On reconstruction of the potential energy from phases of the scattered waves, (Russian) Dokl. Akad. Nauk SSSR (N.S.), 104 (1955), 695-698.

    [14]

    V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.doi: 10.1007/978-3-0348-5485-6.

    [15]

    R. Pike and P. Sabatier, Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, San Diego, 2002.

    [16]

    D. L. Pursey and T. Weber, Formulations of certain Gel'fand-Levitan and Marchenko equations, Phys. Rev. A, 50 (1994), 4472-4477.doi: 10.1103/PhysRevA.50.4472.

    [17]

    W. Rundell and P. Sacks, On the determination of potentials without bound state data, J. Comput. Appl. Math., 55 (1994), 325-347.doi: 10.1016/0377-0427(94)90037-X.

    [18]

    G. Wei and H. K. Xu, Inverse spectral problem with partial information given on the potential and norming constants, Trans. Amer. Math. Soc., 364 (2012), 3265-3288.doi: 10.1090/S0002-9947-2011-05545-5.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(120) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return