-
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
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 |
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
[Back to Top]