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]

Phys. Rev. A, 22 (1980), 1333-1340. doi: 10.1103/PhysRevA.22.1333.  Google Scholar

[2]

SIAM J. Appl. Math., 56 (1996), 219-231. doi: 10.1137/S0036139994273995.  Google Scholar

[3]

J. Math. Anal. Appl., 270 (2002), 247-266. doi: 10.1016/S0022-247X(02)00070-7.  Google Scholar

[4]

Inverse Problems, 22 (2006), 89-114. doi: 10.1088/0266-5611/22/1/006.  Google Scholar

[5]

Izvestiya Akad. Nauk SSSR. Ser. Mat., 15 (1951), 309-360. [In Russian.]  Google Scholar

[6]

Helv. Phys. Acta, 70 (1997), 66-71.  Google Scholar

[7]

Trans. Amer. Math. Soc., 352 (2000), 2765-2787. doi: 10.1090/S0002-9947-99-02544-1.  Google Scholar

[8]

SIAM J. Appl. Math., 55 (1995), 242-254. doi: 10.1137/S0036139993254656.  Google Scholar

[9]

J. Math. Phys., 33 (1992), 3813-3821. doi: 10.1063/1.529990.  Google Scholar

[10]

(Russian) Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), 185-199, 215-216.  Google Scholar

[11]

VNU Science Press, Utrecht, 1978. Google Scholar

[12]

Seismol., 18 (1985), 176-184; N. N. Novikova and V. M. Markushevich, Comput. Seismol., 18 (1987), 164-172. [English Translation.] Google Scholar

[13]

(Russian) Dokl. Akad. Nauk SSSR (N.S.), 104 (1955), 695-698.  Google Scholar

[14]

Birkhauser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.  Google Scholar

[15]

Academic Press, San Diego, 2002.  Google Scholar

[16]

Phys. Rev. A, 50 (1994), 4472-4477. doi: 10.1103/PhysRevA.50.4472.  Google Scholar

[17]

J. Comput. Appl. Math., 55 (1994), 325-347. doi: 10.1016/0377-0427(94)90037-X.  Google Scholar

[18]

Trans. Amer. Math. Soc., 364 (2012), 3265-3288. doi: 10.1090/S0002-9947-2011-05545-5.  Google Scholar

show all references

References:
[1]

Phys. Rev. A, 22 (1980), 1333-1340. doi: 10.1103/PhysRevA.22.1333.  Google Scholar

[2]

SIAM J. Appl. Math., 56 (1996), 219-231. doi: 10.1137/S0036139994273995.  Google Scholar

[3]

J. Math. Anal. Appl., 270 (2002), 247-266. doi: 10.1016/S0022-247X(02)00070-7.  Google Scholar

[4]

Inverse Problems, 22 (2006), 89-114. doi: 10.1088/0266-5611/22/1/006.  Google Scholar

[5]

Izvestiya Akad. Nauk SSSR. Ser. Mat., 15 (1951), 309-360. [In Russian.]  Google Scholar

[6]

Helv. Phys. Acta, 70 (1997), 66-71.  Google Scholar

[7]

Trans. Amer. Math. Soc., 352 (2000), 2765-2787. doi: 10.1090/S0002-9947-99-02544-1.  Google Scholar

[8]

SIAM J. Appl. Math., 55 (1995), 242-254. doi: 10.1137/S0036139993254656.  Google Scholar

[9]

J. Math. Phys., 33 (1992), 3813-3821. doi: 10.1063/1.529990.  Google Scholar

[10]

(Russian) Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), 185-199, 215-216.  Google Scholar

[11]

VNU Science Press, Utrecht, 1978. Google Scholar

[12]

Seismol., 18 (1985), 176-184; N. N. Novikova and V. M. Markushevich, Comput. Seismol., 18 (1987), 164-172. [English Translation.] Google Scholar

[13]

(Russian) Dokl. Akad. Nauk SSSR (N.S.), 104 (1955), 695-698.  Google Scholar

[14]

Birkhauser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.  Google Scholar

[15]

Academic Press, San Diego, 2002.  Google Scholar

[16]

Phys. Rev. A, 50 (1994), 4472-4477. doi: 10.1103/PhysRevA.50.4472.  Google Scholar

[17]

J. Comput. Appl. Math., 55 (1994), 325-347. doi: 10.1016/0377-0427(94)90037-X.  Google Scholar

[18]

Trans. Amer. Math. Soc., 364 (2012), 3265-3288. doi: 10.1090/S0002-9947-2011-05545-5.  Google Scholar

[1]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[2]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[3]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004

[4]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

[5]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

[6]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[7]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[8]

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021026

[9]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

[10]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

[11]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[12]

Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008

[13]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[14]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002

[15]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[16]

Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021073

[17]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[18]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[19]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[20]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]