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Funktsional. Anal. i Prilozhen., 2010, Volume 44, Issue 4, Pages 34–53 (Mi faa3022)  

This article is cited in 24 scientific papers (total in 24 papers)

Inverse Problems for Sturm–Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability

A. M. Savchuk, A. A. Shkalikov

M. V. Lomonosov Moscow State University

Abstract: Two inverse problems for the Sturm–Liouville operator $Ly=-y"+q(x)y$ on the interval $[0,\pi]$ are studied. For $\theta\ge0$, there is a mapping $F\colon W^{\theta}_2 \to l^{\theta}_B$, $F(\sigma)=\{s_k\}_1^\infty$, related to the first of these problems, where $W^\theta_2= W^{\theta}_2[0,\pi]$ is the Sobolev space, $\sigma =\int q$ is a primitive of the potential $q$, and $l^{\theta}_B$ is a specially constructed finite-dimensional extension of the weighted space $l^{\theta}_2$, where we place the regularized spectral data ${\mathbf s}=\{s_k\}_1^\infty$ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for $\|\sigma - \sigma_1\|_\theta$ via the $l^{\theta}_B$-norm $\|{\mathbf s}-{\mathbf s}_1\|_\theta$ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator $L$ generated by the Dirichlet boundary conditions. The result is new even for the classical case $q\in L_2$, which corresponds to $\theta =1$.

Keywords: inverse Sturm–Liouville problem, singular potentials, stability for inverse problems.

DOI: https://doi.org/10.4213/faa3022

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English version:
Functional Analysis and Its Applications, 2010, 44:4, 270–285

Bibliographic databases:

UDC: 517.984.54
Received: 17.05.2010

Citation: A. M. Savchuk, A. A. Shkalikov, “Inverse Problems for Sturm–Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability”, Funktsional. Anal. i Prilozhen., 44:4 (2010), 34–53; Funct. Anal. Appl., 44:4 (2010), 270–285

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Hryniv R.O., “Analyticity and uniform stability in the inverse spectral problem for Dirac operators”, J. Math. Phys., 52:6 (2011), 063513, 17 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Hryniv R.O., “Analyticity and uniform stability in the inverse singular Sturm-Liouville spectral problem”, Inverse Problems, 27:6 (2011), 065011, 25 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Sadovnichii V.A. Sultanaev Ya.T. Akhtyamov A.M., “Generalization of B. M. Levitan and M. G. Gasymov's solvability theorems to the case of indecomposable boundary conditions”, Dokl. Math., 85:2 (2012), 289–291  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. A. M. Akhtyamov, V. A. Sadovnichy, Ya. T. Sultanaev, “Generalizations of Borg's uniqueness theorem to the case of nonseparated boundary conditions”, Eurasian Math. J., 3:4 (2012), 10–22  mathnet  mathscinet  zmath
    5. A. M. Savchuk, A. A. Shkalikov, “On the Interpolation of Analytic Mappings”, Math. Notes, 94:4 (2013), 547–550  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. A. M. Savchuk, A. A. Shkalikov, “Uniform stability of the inverse Sturm–Liouville problem with respect to the spectral function in the scale of Sobolev spaces”, Proc. Steklov Inst. Math., 283 (2013), 181–196  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    8. Eckhardt J. Gesztesy F. Nichols R. Teschl G., “Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials”, J. Lond. Math. Soc.-Second Ser., 88:3 (2013), 801–828  crossref  mathscinet  zmath  isi  scopus
    9. Eckhardt J., Teschl G., “Sturm-Liouville Operators with Measure-Valued Coefficients”, J. Anal. Math., 120 (2013), 151–224  crossref  mathscinet  zmath  isi  elib  scopus
    10. J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, “Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials”, J. Spectr. Theory, 4:4 (2014), 715–768  crossref  mathscinet  zmath  isi  scopus
    11. A.D. Baev, M.B. Zvereva, S.A. Shabrov, “Stieltjes differential in nonlinear momentum problems”, Dokl. Math., 90:2 (2014), 613–615  crossref  crossref  mathscinet  zmath  zmath  isi  elib  scopus
    12. I.M. Nabiev, “Determination of the diffusion operator on an interval”, Colloq. Math., 134:2 (2014), 165–178  crossref  mathscinet  zmath  isi  scopus
    13. V. E. Vladikina, A. A. Shkalikov, “Asymptotics of the Solutions of the Sturm–Liouville Equation with Singular Coefficients”, Math. Notes, 98:6 (2015), 891–899  mathnet  crossref  crossref  mathscinet  isi  elib
    14. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M., “Solvability Theorems For An Inverse Nonself-Adjoint Sturm-Liouville Problem With Nonseparated Boundary Conditions”, Differ. Equ., 51:6 (2015), 717–725  crossref  mathscinet  zmath  isi  scopus
    15. A. M. Savchuk, “Reconstruction of the Potential of the Sturm–Liouville Operator from a Finite Set of Eigenvalues and Normalizing Constants”, Math. Notes, 99:5 (2016), 715–728  mathnet  crossref  crossref  mathscinet  isi  elib
    16. Koshmanenko V. Dudkin M., “Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators”, Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators, Operator Theory Advances and Applications, 253, Springer Int Publishing Ag, 2016, 1–237  crossref  mathscinet  isi
    17. A. A. Aitbaeva, A. M. Akhtyamov, “Identification of the fixedness and loadedness of an end of an Euler–Bernoulli beam from its natural vibration frequencies”, J. Appl. Industr. Math., 11:1 (2017), 1–7  mathnet  crossref  crossref  mathscinet  elib
    18. A. M. Akhtyamov, V. A. Sadovnichy, Ya. T. Sultanaev, “Inverse problem for the diffusion operator with symmetric functions and general boundary conditions”, Eurasian Math. J., 8:1 (2017), 10–22  mathnet
    19. R. Ch. Kulaev, A. B. Shabat, “Some properties of Jost functions for Schrödinger equation with distribution potential”, Ufa Math. J., 9:4 (2017), 59–71  mathnet  crossref  isi  elib
    20. V. A. Sadovnichii, Ya. T. Sultanaev, A. M. Akhtyamov, “Inverse problem for a differential operator with nonseparated boundary conditions”, Dokl. Math., 97:2 (2018), 181–183  mathnet  crossref  crossref  mathscinet  zmath  isi  scopus
    21. Sadovnichy V.A., Sultanaev Ya.T., Aklityamov A.M., “On the Uniqueness of the Solution of the Inverse Sturm-Liouville Problem With Nonseparated Boundary Conditions on a Geometric Graph”, Dokl. Math., 98:1 (2018), 338–340  crossref  mathscinet  zmath  isi  scopus
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  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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