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

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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

Abstract
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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.

Keywords: inverse problem, Jost function, Marchenko equation., Gel'fand-Levitan integral equation, Bargmann-Jost-Kohn class, scattering data.

Mathematics Subject Classification: Primary: 34A55; Secondary: 34L40, 34L2.

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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	B. M. Levitan, Inverse Sturm-Liouville Problems,, VNU Science Press, (1978). Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	V. A. Marchenko, Sturm-Liouville Operators and Applications,, Birkhauser, (1986). doi: 10.1007/978-3-0348-5485-6. Google Scholar
[15]	R. Pike and P. Sabatier, Scattering and Inverse Scattering in Pure and Applied Science,, Academic Press, (2002). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	B. M. Levitan, Inverse Sturm-Liouville Problems,, VNU Science Press, (1978). Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	V. A. Marchenko, Sturm-Liouville Operators and Applications,, Birkhauser, (1986). doi: 10.1007/978-3-0348-5485-6. Google Scholar
[15]	R. Pike and P. Sabatier, Scattering and Inverse Scattering in Pure and Applied Science,, Academic Press, (2002). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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