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Stability of the Reconstruction Discontinuous Sturm-Liouville Problem

Year 2019, Volume: 68 Issue: 1, 484 - 499, 01.02.2019

Ahu Ercan Etibar Panakhov

https://doi.org/10.31801/cfsuasmas.430861
Cited By: 2

Abstract

In this work, we study stability of the inverse spectral problem for the Sturm-Liouville operator -D²+q with discontinuity boundary conditions inside a finite closed interval. We use the method which is given by Ryabushko for regular Sturm-Liouville operator in <cite>ryb</cite> to obtain stability results. These results give a bound for the difference between the spectral functions of associated problems. In addition, we give asymptotic representation of the eigenvalues and a formula for the representation of the norming constants by two spectra.

Keywords

Discontinuity stability spectral function Sturm-Liouville

References

  • Levitan, B. M., On the determination of the Sturm-Liouville operator from one and two spectra, Math. Ussr, Izvestija, 12, (1978), 179-193.
  • Levitan, B. M., Inverse Sturm-Liouville problems, Nauka, Moscow, 1984.
  • Levitan, B. M. and Gasymov, M. G., Determination of a differential equations by its two spectra, Russian Math Surveys, 19, (1964), 1-63.
  • Levitan, B. M. and Sargsjan, I. S., Introduction to spectral theory, American Mathematical Society, Providence, RI, USA, 1975.
  • Panakhov, E. S. and Sat, M., Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential, Bound. Value Probl., 49, (2013), 1-9.
  • Borg, G., Eine Umkehrung der Sturm-Liouvilleschen eigenwertaufgabe, Acta Math., 78, (1945), 1-96.
  • Hochstadt, H., The inverse Sturm-Liouville problem, Comm. On Pure and Applied Mathematics, XXVI, (1973), 715-729.
  • Gelfand, I. M. and Levitan, B. M., On the determination of a differantial equation from its spectral function, Amer. Math. Soc. Transl., 1, (1955), 253-304.
  • Albeverio, S., Hryniv, R. O. and Mykytyuk, Y., Inverse spectral problems for coupled oscillating Reconstruction from three spectra, Methods Funct. Anal. Topology, 13 (2007), 110-123.
  • Marchenko, V. A., Certain problems of the theory of one dimensional linear differential operators of the second order, Trudy Moskov. Mat. Obsc., 1, (1952), 327-420.
  • Hald, O. H., Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37, (1984), 539-577.
  • Sen, E. and Mukhtarov, O. S., Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions, Mediterr. J. Math., 13(1), (2016), 153-170.
  • Aydemir, K. and Muhtaroğlu, O., Asymptotic distribution of eigenvalues and eigenfunctions for a multi point discontinuous Sturm-Liouville problem, Electron. J. Differential Equations, 131, (2016), 1-12.
  • Manafov, M. Dzh., Inverse spectral problems for energy-dependent Sturm-Liouville equations with finitely many point Delta-Interactions, Electron. J. Differential Equations, 11, (2016), 1-14.
  • Yurko, V. A., On boundary value problems with jump conditions inside the interval, Differ. Equ., 8, (2000) 1266-1269.
  • Yang, C.-F. and Yang, X.-P., An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions, Appl. Math. Lett., 22, (2009), 1315-1319.
  • Shieh, C. T. and Yurko, V. A., Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347, (2008), 266-272.
  • Amirov, R. Kh., On Sturm--Liouville operators with discontinuity conditions inside an interval, J. Math. Anal. Appl., 317, (2006), 163-176.
  • McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4, (1988), 529--40.
  • Savchuk, A. M. and Shkalikov, A.A., Inverse problems for Sturm--Liouville operators with potentials in Sobolev spaces: Uniform stability, Funkts. Anal. Prilozh., 44, (2010), 34--53 [Funct. Anal. Appl. 44, 270--285].
  • Hryniv, R. O., Analyticity and uniform stability in the inverse singular Sturm--Liouville spectral problem, Inverse Problems, 27, (2011), 065011.
  • Ryabushko, T. I., Stability of the reconstruction of a Sturm-Liouville operator from two spectra, II. Teor. Funksts. Anal., Prilozhen., 18, (1973), 176-85 (in Russian).
  • Marchenko, V. A. and Maslov, K. V., Stability of the problem of reconstruction of the Sturm-Liouville operator in terms of the spectral function, Mathematics of the USSR Sbornik, 81, (1970), 525-51 (in Russian).
  • Panakhov, E. S. and Ercan, A., Stability problem for singular Sturm-Liouville equation, TWMS J. Pure Appl. Math., 8(2), (2017), 148-159.
  • Ercan, A. and Panakhov, E. S., Stability problem for singular Dirac equation system on finite interval, AIP Conf. Proc., 1798, 020054; doi: 10.1063/1.4972646, (2017), 1-9.

Year 2019, Volume: 68 Issue: 1, 484 - 499, 01.02.2019

Ahu Ercan Etibar Panakhov

https://doi.org/10.31801/cfsuasmas.430861
Cited By: 2

Abstract

References

  • Levitan, B. M., On the determination of the Sturm-Liouville operator from one and two spectra, Math. Ussr, Izvestija, 12, (1978), 179-193.
  • Levitan, B. M., Inverse Sturm-Liouville problems, Nauka, Moscow, 1984.
  • Levitan, B. M. and Gasymov, M. G., Determination of a differential equations by its two spectra, Russian Math Surveys, 19, (1964), 1-63.
  • Levitan, B. M. and Sargsjan, I. S., Introduction to spectral theory, American Mathematical Society, Providence, RI, USA, 1975.
  • Panakhov, E. S. and Sat, M., Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential, Bound. Value Probl., 49, (2013), 1-9.
  • Borg, G., Eine Umkehrung der Sturm-Liouvilleschen eigenwertaufgabe, Acta Math., 78, (1945), 1-96.
  • Hochstadt, H., The inverse Sturm-Liouville problem, Comm. On Pure and Applied Mathematics, XXVI, (1973), 715-729.
  • Gelfand, I. M. and Levitan, B. M., On the determination of a differantial equation from its spectral function, Amer. Math. Soc. Transl., 1, (1955), 253-304.
  • Albeverio, S., Hryniv, R. O. and Mykytyuk, Y., Inverse spectral problems for coupled oscillating Reconstruction from three spectra, Methods Funct. Anal. Topology, 13 (2007), 110-123.
  • Marchenko, V. A., Certain problems of the theory of one dimensional linear differential operators of the second order, Trudy Moskov. Mat. Obsc., 1, (1952), 327-420.
  • Hald, O. H., Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37, (1984), 539-577.
  • Sen, E. and Mukhtarov, O. S., Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions, Mediterr. J. Math., 13(1), (2016), 153-170.
  • Aydemir, K. and Muhtaroğlu, O., Asymptotic distribution of eigenvalues and eigenfunctions for a multi point discontinuous Sturm-Liouville problem, Electron. J. Differential Equations, 131, (2016), 1-12.
  • Manafov, M. Dzh., Inverse spectral problems for energy-dependent Sturm-Liouville equations with finitely many point Delta-Interactions, Electron. J. Differential Equations, 11, (2016), 1-14.
  • Yurko, V. A., On boundary value problems with jump conditions inside the interval, Differ. Equ., 8, (2000) 1266-1269.
  • Yang, C.-F. and Yang, X.-P., An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions, Appl. Math. Lett., 22, (2009), 1315-1319.
  • Shieh, C. T. and Yurko, V. A., Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347, (2008), 266-272.
  • Amirov, R. Kh., On Sturm--Liouville operators with discontinuity conditions inside an interval, J. Math. Anal. Appl., 317, (2006), 163-176.
  • McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4, (1988), 529--40.
  • Savchuk, A. M. and Shkalikov, A.A., Inverse problems for Sturm--Liouville operators with potentials in Sobolev spaces: Uniform stability, Funkts. Anal. Prilozh., 44, (2010), 34--53 [Funct. Anal. Appl. 44, 270--285].
  • Hryniv, R. O., Analyticity and uniform stability in the inverse singular Sturm--Liouville spectral problem, Inverse Problems, 27, (2011), 065011.
  • Ryabushko, T. I., Stability of the reconstruction of a Sturm-Liouville operator from two spectra, II. Teor. Funksts. Anal., Prilozhen., 18, (1973), 176-85 (in Russian).
  • Marchenko, V. A. and Maslov, K. V., Stability of the problem of reconstruction of the Sturm-Liouville operator in terms of the spectral function, Mathematics of the USSR Sbornik, 81, (1970), 525-51 (in Russian).
  • Panakhov, E. S. and Ercan, A., Stability problem for singular Sturm-Liouville equation, TWMS J. Pure Appl. Math., 8(2), (2017), 148-159.
  • Ercan, A. and Panakhov, E. S., Stability problem for singular Dirac equation system on finite interval, AIP Conf. Proc., 1798, 020054; doi: 10.1063/1.4972646, (2017), 1-9.
There are 25 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Ahu Ercan 0000-0001-6290-2155

Etibar Panakhov 0000-0002-5309-048X

Publication Date February 1, 2019
Submission Date December 22, 2017
Acceptance Date February 16, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Ercan, A., & Panakhov, E. (2019). Stability of the Reconstruction Discontinuous Sturm-Liouville Problem. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 484-499. https://doi.org/10.31801/cfsuasmas.430861
AMA Ercan A, Panakhov E. Stability of the Reconstruction Discontinuous Sturm-Liouville Problem. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):484-499. doi:10.31801/cfsuasmas.430861
Chicago Ercan, Ahu, and Etibar Panakhov. “Stability of the Reconstruction Discontinuous Sturm-Liouville Problem”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 484-99. https://doi.org/10.31801/cfsuasmas.430861.
EndNote Ercan A, Panakhov E (February 1, 2019) Stability of the Reconstruction Discontinuous Sturm-Liouville Problem. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 484–499.
IEEE A. Ercan and E. Panakhov, “Stability of the Reconstruction Discontinuous Sturm-Liouville Problem”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 484–499, 2019, doi: 10.31801/cfsuasmas.430861.
ISNAD Ercan, Ahu - Panakhov, Etibar. “Stability of the Reconstruction Discontinuous Sturm-Liouville Problem”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 484-499. https://doi.org/10.31801/cfsuasmas.430861.
JAMA Ercan A, Panakhov E. Stability of the Reconstruction Discontinuous Sturm-Liouville Problem. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:484–499.
MLA Ercan, Ahu and Etibar Panakhov. “Stability of the Reconstruction Discontinuous Sturm-Liouville Problem”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 484-99, doi:10.31801/cfsuasmas.430861.
Vancouver Ercan A, Panakhov E. Stability of the Reconstruction Discontinuous Sturm-Liouville Problem. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):484-99.

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