American Institute of Mathematical Sciences

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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

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

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
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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

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