# American Institute of Mathematical Sciences

February  2015, 9(1): 239-255. doi: 10.3934/ipi.2015.9.239

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

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaaxi 710062, China 2 Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

Received  March 2013 Revised  July 2014 Published  January 2015

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.
Citation: Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239
##### References:
 [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. 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. 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. 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. 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. [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. [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. 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. 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. 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. [11] B. M. Levitan, Inverse Sturm-Liouville Problems,, VNU Science Press, (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. [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. [14] V. A. Marchenko, Sturm-Liouville Operators and Applications,, Birkhauser, (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, (2002). [16] D. L. Pursey and T. Weber, Formulations of certain Gel'fand-Levitan and Marchenko equations,, Phys. Rev. A, 50 (1994), 4472. 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. 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. doi: 10.1090/S0002-9947-2011-05545-5.

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##### References:
 [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. 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. 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. 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. 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. [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. [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. 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. 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. 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. [11] B. M. Levitan, Inverse Sturm-Liouville Problems,, VNU Science Press, (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. [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. [14] V. A. Marchenko, Sturm-Liouville Operators and Applications,, Birkhauser, (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, (2002). [16] D. L. Pursey and T. Weber, Formulations of certain Gel'fand-Levitan and Marchenko equations,, Phys. Rev. A, 50 (1994), 4472. 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. 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. doi: 10.1090/S0002-9947-2011-05545-5.
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