Taiwanese Journal of Mathematics

Recovery of the Schrödinger Operator on the Half-line from a Particular Set of Eigenvalues

Xiao-Chuan Xu and Chuan-Fu Yang

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In this work, we study the Schrödinger operator on the half-line with a selfadjoint boundary condition. We prove that a particular set of eigenvalues can uniquely determine the potential. The reconstruction algorithm for recovering the potential from the particular data is provided.

Article information

Taiwanese J. Math., Volume 21, Number 6 (2017), 1325-1334.

Received: 20 June 2016
Revised: 24 February 2017
Accepted: 12 March 2017
First available in Project Euclid: 17 August 2017

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Primary: 34A55: Inverse problems 34B24: Sturm-Liouville theory [See also 34Lxx] 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)

Schrödinger operator on the half-line inverse eigenvalue problem reconstruction algorithm


Xu, Xiao-Chuan; Yang, Chuan-Fu. Recovery of the Schrödinger Operator on the Half-line from a Particular Set of Eigenvalues. Taiwanese J. Math. 21 (2017), no. 6, 1325--1334. doi:10.11650/tjm/8026. https://projecteuclid.org/euclid.twjm/1502935249

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  • T. Aktosun, Inverse scattering for vowel articulation with frequency-domain data, Inverse Problems 21 (2005), no. 3, 899–914. https://doi.org/10.1088/0266-5611/21/3/007
  • T. Aktosun, P. Sacks and M. Unlu, Inverse problems for selfadjoint Schrödinger operators on the half line with compactly supported potentials, J. Math. Phys. 56 (2015), no. 2, 022106, 33 pp. https://doi.org/10.1063/1.4907558
  • T. Aktosun and R. Weder, Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation, Inverse Problems 22 (2006), no. 1, 89–114. https://doi.org/10.1088/0266-5611/22/1/006
  • B. J. Forbes, E. R. Pike and D. B. Sharp, The acoustical Klein-Gordon equation: The wave-mechanical step and barrier potential functions, J. Acoust. Soc. Am. 114 (2003), no. 3, 1291–1302. https://doi.org/10.1121/1.1590314
  • G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, Huntington, NY, 2001.
  • F. Gesztesy and B. Simon, A new approach to inverse spectral theory, II: General real potentials and the connection to the spectral measure, Ann. of Math. 152 (2000), no. 2, 593–643. https://doi.org/10.2307/2661393
  • E. Korotyaev, Inverse resonance scattering on the half line, Asymptot. Anal. 37 (2004), no. 3-4, 215–226.
  • V. A. Marchenko, Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applictions 22, Birkhäuser Verlag, Basel, 1986. https://doi.org/10.1007/978-3-0348-5485-6
  • J. R. McLaughlin and W. Rundell, A uniqueness theorem for an inverse Sturm-Liouville problem, J. Math. Phys. 28 (1987), no. 7, 1471–1472. https://doi.org/10.1063/1.527500
  • A. G. Ramm, Property C for ordinary differential equations and applications to inverse scattering, Z. Anal. Anwendungen 18 (1999), no. 2, 331–348. https://doi.org/10.4171/zaa/885
  • ––––, One-dimensional inverse scattering and spectral problems, Cubo 6 (2004), no. 1, 313–426.
  • ––––, Recovery of the potential from $I$-function, Rep. Math. Phys. 74 (2014), no. 2, 135–143. https://doi.org/10.1016/s0034-4877(14)00020-2
  • A. G. Ramm and B. Simon, A new approach to inverse spectral theory, III: Short-range potentials, J. Anal. Math. 80 (2000), no. 1, 319–334. https://doi.org/10.1007/bf02791540
  • X.-C. Xu, C.-F. Yang and H.-Z. You, Inverse spectral analysis for Regge problem with partial information on the potential, Results Math. 71 (2017), no. 3-4, 983–996. https://doi.org/10.1007/s00025-015-0523-6
  • V. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. https://doi.org/10.1515/9783110940961