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Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter

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Abstract

Inverse nodal problem for the Dirac operator on a finite interval with boundary conditions depending polynomially on the spectral parameter is considered. We prove that a set of nodal points of one of the components of the eigenfunctions uniquely determines the coefficients of the Dirac equations and the polynomials of the boundary conditions.

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References

  1. Amirov R.Kh., Keskin B., Ozkan A.S.: Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition. Ukr. Math. J. 61, 1365–1379 (2009)

    Article  MathSciNet  Google Scholar 

  2. Annaby M.H., Tharwat M.M.: On sampling and Dirac systems with eigenparameter in the boundary conditions. J. Appl. Math. Comput. 36, 291–317 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Buterin S.A., Shieh Chung-Tsun: Inverse nodal problem for differential pencils. Appl. Math. Lett. 22, 1240–1247 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buterin S.A., Shieh Chung-Tsun: Incomplete inverse spectral and nodal problems for differential pencils. Results Math. 62, 167–179 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen Y.T., Cheng Y.H., Law C.K., Tsay J.: L 1 convergence of the reconstruction formula for the potential function. Proc. Am. Math. Soc. 130, 2319–2324 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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  MATH  MathSciNet  Google Scholar 

  7. Collatz L.: Eigenwertaufgaben mit Technischen Anwendungen. Akad. Verlagsgesellschaft Geest Portig, Leipzig (1963)

  8. Freiling, G., Yurko, V.A.: Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter. Inverse Probl. 26, 17 (2010)

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Gasymov, M.G., Dzhabiev, T.T.: Determination of a system of Dirac differential equations using two spectra. In: Proceedings of School-Seminar on the Spectral Theory of Operators and Representations of Group Theory (in Russian). Élm, Baku, pp. 46–71 (1975)

  11. Gasymov M.G.: Inverse problem of the scattering theory for Dirac system of order 2n. Tr. Mosk. Mat. Obshch. 19, 41–112 (1968)

    MathSciNet  Google Scholar 

  12. Guo Y., Wei G.: Inverse problems: dense nodal subset on an interior subinterval. J. Differ. Equ. 255, 2002–2017 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hald O.H., McLaughlin J.R.: Solutions of inverse nodal problems. Inverse Probl. 5, 307–347 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ignatiev M.Yu., Shieh C.-T.: Recovering the Sturm–Liouville operator with singular potential using nodal data. Results Math. 57, 183–194 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kerimov N.B.: A boundary value problem for the Dirac system with a spectral parameter in the boundary conditions. Differ. Equ. 38, 164–174 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kraft R.E., Wells W.R.: Adjointness properties for differential systems with eigenvalue-dependent boundary conditions, with application to flow-duct acoustics. J. Acoust. Soc. Am. 61, 913–922 (1977)

  17. Kuryshova Y.V., Shieh C.-T.: An inverse nodal problem for integro-differential operators. J. Inverse Ill Posed Probl. 18, 357–369 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Levin, B.Ya.: Lectures on Entire Functions. Translation of Mathematical Monograph, vol. 150. American Mathematical Society, Providence, RI (1996)

  19. Levitan, B.M., Sargsyan, I.S.: Sturm–Liouville and Dirac Operators (Russian). Nauka, Moscow (1988) (English translation, Kluwer, Dordrecht, 1991)

  20. Levitan B.M.: Inverse Sturm–Liouville Problems. VNU Science Press, Utrecht (1987)

    MATH  Google Scholar 

  21. Marchenko, V.A.: Sturm–Liouville Operators and Their Applications. Naukova Dumka, Kiev (1977) (English translation, Birkhäuser, Basel, 1986)

  22. McLaughlin J.R.: Analytical methods for recovering coefficients in differential equations from spectral data. SIAM Rev. 28, 53–72 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Walter J.: Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133, 301–312 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yang X.F.: A new inverse nodal problem. J. Differ. Equ. 169, 633–653 (2001)

    Article  MATH  Google Scholar 

  27. Yang C.-F., Huang Z.-Y.: Reconstruction of the Dirac operator from nodal data. Integr. Equ. Oper. Theory 66, 539–551 (2010)

    Article  MATH  Google Scholar 

  28. Yang C.-F., Pivovarchik V.N.: Inverse nodal problem for Dirac system with spectral parameter in boundary conditions. Complex Anal. Oper. Theory 7, 1211–1230 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yurko, V.A.: Inverse Spectral Problems for Differential Operators and Their Applications. Gordon and Breach, Amsterdam (2000)

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Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

    Yongxia Guo & Guangsheng Wei

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  1. Yongxia Guo
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  2. Guangsheng Wei
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Correspondence to Yongxia Guo.

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Guo, Y., Wei, G. Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter. Results. Math. 67, 95–110 (2015). https://doi.org/10.1007/s00025-014-0396-0

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