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

Previous Article
A new spectral method for $l_1$-regularized minimization
IPI Home
This Issue
Next Article
Sparse signals recovery from noisy measurements by orthogonal matching pursuit

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

Guangsheng Wei ^1, and Hong-Kun Xu ^2,

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
Related Papers

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 and 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), 1333-1340. 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.

show all references

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), 1333-1340. 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.

[1]	Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[2]	John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[3]	Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems and Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[4]	Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[5]	Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021060
[6]	Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[7]	Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73
[8]	Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[9]	Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435
[10]	Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations and Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[11]	Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[12]	Wei-Kang Xun, Shou-Fu Tian, Tian-Tian Zhang. Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021259
[13]	Yuan Li, Shou-Fu Tian. Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation. Communications on Pure and Applied Analysis, 2022, 21 (1) : 293-313. doi: 10.3934/cpaa.2021178
[14]	Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[15]	Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[16]	Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77
[17]	Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[18]	Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055
[19]	Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[20]	Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Guangsheng Wei Hong-Kun Xu
on Google Scholar
Guangsheng Wei Hong-Kun Xu

[Back to Top]