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Uniqueness Theorems for Sturm–Liouville Operators with Boundary Conditions Polynomially Dependent on the Eigenparameter from Spectral Data

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Abstract

In this paper, we discuss the inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter. We establish some uniqueness theorems on the potential q(x) for the half inverse problem and the interior inverse problem from spectral data, respectively.

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References

  1. Chernozhukova A.Y., Freiling G.: A uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter. Inverse Probl. Sci. Eng. 17, 777–785 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Freiling G., Yurko V.A.: Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter. Inverse Probl. 26(6), 055003 (2010)

    Article  MathSciNet  Google Scholar 

  3. Yang C.F., Yang X.P.: Inverse nodal problem for the Sturm–Liouville equation with polynomially dependent on the eigenparameter. Inverse Prob. Sci. Eng. 19, 951–961 (2011)

    Article  MATH  Google Scholar 

  4. Fulton C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh 77a, 293–308 (1977)

    Article  MathSciNet  Google Scholar 

  5. Binding P.A., Browne P.J., Seddighi K.: Sturm–Liouville problems with eigenparameter dependent boundary conditions. Proc. Roy. Soc. Edinburgh. 37, 57–72 (1993)

    MathSciNet  Google Scholar 

  6. Browne P.J., Sleeman B.D.: Inverse nodal problems for Sturm–Liouville equations with eigenparameter dependent boundary conditions. Inverse Probl. 12, 377–381 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guliyev N.J.: The regularized trace formula for the Sturm–Liouville equation with spectral parameter in the boundary conditions. Proc. Inst. Math. Natl. Alad. Sci. Azerb. 22, 99–102 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Gaskell R.E.: A problem in heat conduction and an expansion theorem. Am. J. Math. 64, 447–455 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bauer W.F.: Modified Sturm–Liouville systems. Quart. Appl. Math. 11, 273–282 (1953)

    MathSciNet  MATH  Google Scholar 

  10. Wang Y.P.: An interior inverse problem for Sturm–Liouville operators with eigenparameter dependent boundary conditions. TamKang J. Math. 42(3), 395–403 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Shieh C.T., Buterin S.A., Ignatiev M.: On Hochstadt–Liberman theorem for Sturm–Liouville operators. Far East J. Appl. Math. 52(2), 131–146 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Hochstadt H., Lieberman B.: An inverse Sturm–Liouville problem wity mixed given data. SIAM J. Appl. Math. 34, 676–680 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Castillo R.D.R.: On boundary conditions of an inverse Sturm–Liouville problem. SIAM. J. APPL. MATH. 50(6), 1745–1751 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Suzuki, T.: Inverse problems for heat equations on compact intervals and on circles. I. J. Math. Soc. Japan 38, 39–65, MR 87f:35241 (1986)

    Google Scholar 

  15. Sakhnovich L.: Half inverse problems on the finite interval. Inverse Probl. 17, 527–532 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hryniv R., Mykytyuk O.: Half inverse spectral problems for Sturm–Liouville operators with singular potentials. Inverse Probl. 20(5), 1423–1444 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Koyunbakan H., Panakhov E.S.: Half inverse problem for diffusion operators on the finite interval. J. Math. Anal. Appl. 326, 1024–1030 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Buterin S.A.: On half inverse problem for differential pencils with the spectral parameter in boundary conditions. Tamkang J. Math. 42(3), 355–364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wei G.S., Xu H.K.: On the missing eigenvalue problem for an inverse Sturm–Liouville problem. J. Math. Pure Appl. 91, 468–475 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Gesztesy F., Simon B.: Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum. Trans. Am. Math. Soc. 352, 2765–2787 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mochizuki K., Trooshin I.: Inverse problem for interior spectral data of Sturm–Liouville operator. J. Inverse Ill-Posed Probl. 9, 425–433 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Yang C.F., Yang X.P.: An interior inverse problem for the Sturm–Liouville operator with discontinuous conditions. Appl. Math. Lett. 22, 1315–1319 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hald O.H.: The Sturm–Liouville problem with symmetric potentials. Acta. Math. 141, 262–291 (1978)

    Article  MathSciNet  Google Scholar 

  24. McLaughlin J.R.: Inverse spectral theory using nodal points as data-a uniqueness result. J. Differ. Equ. 73, 354–362 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shieh C.T., Yurko V.A.: Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J. Math. Anal. Appl. 347(1), 266–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Law C.K., Tsay J.: On the well-posedness of the inverse nodal problem. Inverse Probl. 17, 1493–1512 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen C.L., Shieh C.T.: An inverse nodal problem for vectorial Sturm–Liouville equation. Inverse Probl. 16, 349–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ozkan A.S., Amirovan R.Kh.: Interior inverse problem for the impulsive Dirac operator. Tamkang J. Math. 42(3), 259–263 (2011)

    Article  MathSciNet  Google Scholar 

  29. Levin, B.J.: Distribution of zeros of entire functions, AMS. Transl, vol. 5. Providence (1964)

  30. Yurko, V.A.: Method of spectral mappings in the inverse problem theory, VSP, Utrecht: Inverse Ill-posed Problems Ser (2002)

  31. Freiling G., Yurko V.A.: Inverse Sturm–Liouville problems and their applications. Nova Science Publishers, Huntington (2001)

    MATH  Google Scholar 

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  1. Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037, Jiangsu, People’s Republic of China

    Yu Ping Wang

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  1. Yu Ping Wang
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Wang, Y.P. Uniqueness Theorems for Sturm–Liouville Operators with Boundary Conditions Polynomially Dependent on the Eigenparameter from Spectral Data. Results. Math. 63, 1131–1144 (2013). https://doi.org/10.1007/s00025-012-0258-6

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  • DOI: https://doi.org/10.1007/s00025-012-0258-6

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