Abstract
An inverse nodal problem lies in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with an inverse nodal problem of reconstructing the Dirac system with the spectral parameter in the boundary conditions. We prove that a set of nodal points of one of the components of the eigenfunctions uniquely determines all the parameters of the boundary conditions and the coefficients of the Dirac equations. We also provide a constructive procedure for solving this inverse nodal problem.
Similar content being viewed by others
References
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. Ukrainian Math. J. 61, 1365–1379 (2009)
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)
Binding P.A., Browne P.J., Watson B.A.: Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. J. Math. Anal. Appl. 291, 246–261 (2004)
Browne P.J., Sleeman B.D.: Inverse nodal problem for Sturm-Liouville equation with eigenparameter dependent boundary conditions. Inverse Probl. 12, 377–381 (1996)
Buterin S.A., Shieh C.T.: Inverse nodal problem for differential pencils. Appl. Math. Lett. 22, 1240–1247 (2009)
Chadan K., Sabatier P.: Inverse problems in quantum scattering theory. Springer, New York (1977)
Cheng Y.H., Law C.K.: On the quasi-nodal map for the Sturm-Liouville problem. Proc. R. Soc. Edinburgh A 136, 71–86 (2006)
Cheng Y.H., Law C.K.: The inverse nodal problem for Hill’s equation. Inverse Probl. 22, 891–901 (2006)
Cheng Y.H., Law C.K., Tsay J.: Remarks on a new inverse nodal problem. J. Math. Anal. Appl. 248, 145–155 (2000)
Currie S., Watson B.A.: Inverse nodal problems for Sturm-Liouville equations on graphs. Inverse Probl. 23, 2029–2040 (2007)
Freiling G., Yurko V.A.: Inverse Sturm-Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001)
Fulton C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinburgh 77(A), 293–308 (1977)
Gasymov M.G.: Inverse problem of scattering theory for Dirac system of order 2n. Tr. Mosk. Mat. Obshch. 19, 41–112 (1968)
Hald O.H., McLaughlin J.R.: Solutions of inverse nodal problems. Inverse Probl. 5, 307–347 (1989)
Kerimov, N. B.: A boundary value problem for the Dirac system with a spectral parameter in the boundary conditions. Diff. Equ. 38(2), 164–174 (2002). (Translated from Differentsial’nye Uravneniya, 2 (2002), 155–164)
Kong Q., Zettl A.: Dependence of eigenvalues of Sturm-Liouville problems on the boundary. J. Diff. Equ. 126, 1–19 (1996)
Koyunbakan H.: Erratum to “Inverse nodal problem for differential operator with eigenvalue in the boundary condition”. Appl. Math. Lett. 22, 792–795 (2009)
Law C.K., Tsay J.: On the well-posedness of the inverse nodal problem. Inverse Probl. 17, 1493–1512 (2001)
Law C.K., Yang C.F.: Reconstructing the potential function and its derivatives using nodal data. Inverse Probl. 14(2), 299–312 (1998)
Levitan, B.M., Sargsjan, I.S.: Sturm-Liouville and Dirac Operators (Russian), Nauka, Moscow (1988). (English transl., Kluwer, Dordrecht, 1991)
McCarthy C.M., Rundell W.: Eigenparameter dependent inverse Sturm-Liouville problems. Numer. Funct. Anal. Optim. 24, 85–105 (2003)
McLaughlin J.R.: Inverse spectral theory using nodal points as data–a uniqueness result. J. Diff. Equ. 73, 354–362 (1988)
Mennicken R., Möller M.: Non-Self-Adjoint Boundary Value Problems. North-Holland Mathematic Studies, vol. 192. Amsterdam, North-Holland (2003)
Pivovarchik V.: Direct and inverse three-point Sturm-Liouville problems with parameter-dependent boundary conditions. Asymptotic Analysis 26, 219–238 (2001)
Pöschel J., Trubowitz E.: Inverse Spectral Theory. Academic Press, Orlando (1987)
Ramm A.G.: Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York (2005)
Rundell W., Sacks P.E.: The reconstruction of Sturm-Liouville operators. Inverse Probl. 8, 457–482 (1992)
Shen C.L., Shieh C.T.: An inverse nodal problem for vectorial Sturm-Liouville equation. Inverse Probl. 16, 349–356 (2000)
van der Mee C., Pivovarchik V.N.: A Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter. Funct. Anal. Appl. 36, 315–317 (2002)
Watson B.A.: Inverse spectral problems for weighted Dirac systems. Inverse Probl. 15, 793–805 (1999)
Yamamoto M.: Inverse eigenvalue problem for a vibration of a string with viscous drag. J. Math. Anal. Appl. 152, 20–34 (1990)
Yang C.F., Huang Z. Y.: Reconstruction of the Dirac operator from nodal data. Integr. Equ. Oper. Theory 66, 539–551 (2010)
Yang X.F.: A solution of the inverse nodal problem. Inverse Probl. 13, 203–213 (1997)
Yurko V.A.: Inverse Spectral Problems for Differential Operators and Their Applications. Gordon and Breach, Amsterdam (2000)
Yurko V.A.: Inverse nodal problems for Sturm-Liouville operators on star-type graphs. J. Inv. Ill Posed Probl. 16, 715–722 (2008)
Yurko V.A., Freiling G.: Inverse nodal problems for differential operators on graphs with a cycle. Tamkang J. Math. 41, 15–54 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Harald Woracek.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s11785-011-0202-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-011-0202-x
Keywords
- Dirac system
- Boundary conditions
- Spectral parameter
- Inverse nodal problem
- Asymptotics of eigenvalue
- Reconstruction formula