10.1007/s40096-021-00383-8

Inverse nodal problem for discontinuous Sturm–Liouville operator by new Prüfer Substitutions

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  • Hikmet Koyunbakan1,
  • Seyfollah Mosazadeh*2,
  1. Department of Mathematics, Faculty of Science, Firat University, Elazig, 23119, TR
  2. Department of Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, 87317-53153, IR

Published in Issue 2021-02-17

How to Cite

Koyunbakan, H., & Mosazadeh, S. (2021). Inverse nodal problem for discontinuous Sturm–Liouville operator by new Prüfer Substitutions. Mathematical Sciences, 15(4 (December 2021). https://doi.org/10.1007/s40096-021-00383-8
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Abstract

Abstract In the present paper, a boundary value problem consisting of a Sturm–Liouville equation with boundary conditions dependent on the eigenparameter and discontinuous conditions inside the interval is investigated. We present new Prüfer substitutions and obtain the asymptotic form of eigenvalues, nodal points and nodal lengths. Then, we prove the uniqueness theorem for the solution of the inverse nodal problem, present a constructive procedure for the potential function by using nodal lengths. Finally, we study Lipschitz stability for the inverse problem.

Keywords

  • Prüfer substitutions,
  • Discontinuous conditions,
  • Nodal points,
  • Inverse nodal problem,
  • Lipschitz stability

References

  1. Browne and Sleeman (1996) Inverse nodal problems for Sturm–Liouville equations with eigen parameter dependent boundary conditions (pp. 377-381) https://doi.org/10.1088/0266-5611/12/4/002
  2. Buterin and Shieh (2012) Incomplete inverse spectral and nodal problems for differential pencils (pp. 167-179) https://doi.org/10.1007/s00025-011-0137-6
  3. Chen et al. (2011) Reconstructing potentials from zeros of one eigenfunction (pp. 4831-4851) https://doi.org/10.1090/S0002-9947-2011-05258-X
  4. Collatz (1963) Akad. Verlagsgesellschaft Geest Portig
  5. Freiling and Yurko (2001) NOVA Science Publishers
  6. Fulton (1977) Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions (pp. 293-308) https://doi.org/10.1017/S030821050002521X
  7. Guliyev (2005) Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions (pp. 1315-1330) https://doi.org/10.1088/0266-5611/21/4/008
  8. Gulsen et al. (2018) Numerical investigation of the inverse nodal problem by Chebyshev interpolation method (pp. 123-136) https://doi.org/10.2298/TSCI170612278G
  9. Guo and Wei (2013) Inverse problems: dense nodal subset on an interior subinterval (pp. 2002-2017) https://doi.org/10.1016/j.jde.2013.06.006
  10. Hald (1984) Discontinuous inverse eigenvalue problems (pp. 539-577) https://doi.org/10.1002/cpa.3160370502
  11. Keskin et al. (2011) Inverse spectral problems for discontinuous Sturm–Liouville operator with eigenvalue dependent boundary conditions A1(60) (pp. 15-25)
  12. Khalili et al. (2017) Half inverse problems for the impulsive operator with eigenvalue-dependent boundary conditions (pp. 1-5)
  13. Koyunbakan (2008) The inverse nodal problem for a differential operator with an eigenvalue in the boundary condition (pp. 1301-1305) https://doi.org/10.1016/j.aml.2008.01.003
  14. Koyunbakan et al. (2018) Inverse nodal problem for a pdocumentclass[12pt]{minimal}
  15. usepackage{amsmath}
  16. usepackage{wasysym}
  17. usepackage{amsfonts}
  18. usepackage{amssymb}
  19. usepackage{amsbsy}
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  22. setlength{oddsidemargin}{-69pt}
  23. begin{document}$$p$$end{document}-Laplacian Sturm–Liouville equation with polynomially boundary condition (pp. 1-9)
  24. Law and Tsay (2001) On the well-posedness of the inverse nodal problem (pp. 1493-1512) https://doi.org/10.1088/0266-5611/17/5/317
  25. Levitan and Sargsjan (1991) Kluwer Academic Publishers
  26. McLaughlin (1988) Inverse spectral theory using nodal points as data, a uniqueness result (pp. 354-362) https://doi.org/10.1016/0022-0396(88)90111-8
  27. Mosazadeh (2020) A new approach to asymptotic formulas for eigenfunctions of discontinuous non-selfadjoint Sturm–Liouville operators https://doi.org/10.1007/s11868-020-00350-2
  28. Neamaty and Khalili (2014) Determination of a differential operator with discontinuity from interior spectral data (pp. 1002-1008) https://doi.org/10.1080/17415977.2013.848436
  29. Ozkan and Keskin (2012) Spectral problems for Sturm–Liouville operator with boundary and jump conditions linearly dependent on the eigen parameter (pp. 799-808) https://doi.org/10.1080/17415977.2011.652957
  30. Ozkan and Keskin (2015) Inverse nodal problems for Sturm–Liouville equation with eigen parameter-dependent boundary and jump conditions (pp. 1306-1312) https://doi.org/10.1080/17415977.2014.991730
  31. Pinasco and Scarola (2015) A nodal inverse problem for second order Sturm–Liouville operators with indefinite weights (pp. 819-830)
  32. Pivovarchik (2001) Direct and inverse three-point Sturm–Liouville problems with parameter-dependent boundary conditions (pp. 219-238)
  33. Poschel and Trubowitz (1987) Academic Press Inc
  34. Qin et al. (2019) Inverse nodal problems for the Sturm–Liouville operator with some nonlocal integral conditions (pp. 111-122) https://doi.org/10.4236/jamp.2019.71010
  35. Sadovnichi et al. (2015) Solvability theorems for an inverse nonself-adjoint Sturm–Liouville problem with nonseparated boundary conditions (pp. 717-725) https://doi.org/10.1134/S0012266115060026
  36. Shahriari et al. (2012) Uniqueness for inverse Sturm–Liouville problems with a finite number of transmission conditions (pp. 19-29) https://doi.org/10.1016/j.jmaa.2012.04.048
  37. Shieh et al. (2011) On Hochstadt–Liebermann theorem for Sturm–Liouville operators (pp. 131-146)
  38. Shieh and Yurko (2008) Inverse nodal and inverse spectral problems for discontinuous boundary value problems (pp. 266-272) https://doi.org/10.1016/j.jmaa.2008.05.097
  39. Tikhonov and Samarskii (1963) Oxford Pergamon Press
  40. Tretter (1996) Nonselfadjoint spectral problems for linear pencils N-λPdocumentclass[12pt]{minimal}
  41. usepackage{amsmath}
  42. usepackage{wasysym}
  43. usepackage{amsfonts}
  44. usepackage{amssymb}
  45. usepackage{amsbsy}
  46. usepackage{mathrsfs}
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  48. setlength{oddsidemargin}{-69pt}
  49. begin{document}$$N-lambda P$$end{document} of ordinary differential operators with λdocumentclass[12pt]{minimal}
  50. usepackage{amsmath}
  51. usepackage{wasysym}
  52. usepackage{amsfonts}
  53. usepackage{amssymb}
  54. usepackage{amsbsy}
  55. usepackage{mathrsfs}
  56. usepackage{upgreek}
  57. setlength{oddsidemargin}{-69pt}
  58. begin{document}$$lambda $$end{document}-linear boundary conditions: completeness results (pp. 222-248) https://doi.org/10.1007/BF01191859
  59. Walter (1973) Regular eigenvalue problems with eigenvalue parameter in the boundary condition (pp. 301-312) https://doi.org/10.1007/BF01177870
  60. Wang and Yurko (2016) On the inverse nodal problems for discontinuous Sturm–Liouville operators (pp. 4086-4109) https://doi.org/10.1016/j.jde.2015.11.004
  61. Willis (1985) Inverse Sturm–Liouville problems with two discontinuities (pp. 263-290) https://doi.org/10.1088/0266-5611/1/3/010
  62. Yang (2013) Inverse nodal problems of discontinuous Sturm–Liouville operator (pp. 1992-2014) https://doi.org/10.1016/j.jde.2012.11.018