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PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION

Year 2017, Volume: 30 Issue: 1, 371 - 379, 14.03.2017

Abstract

In this paper, we determine the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP)

-y′′+q(x)y=λ²y, x∈ℝ₊=[0,∞]


(α₀+α₁λ+α₂λ²)y′(0)-(β₀+β₁λ+β₂λ²)y(0)=0,

where q is a complex-valued function, α_{i}, β_{i}∈ℂ, i=0,1,2 and λ is a eigenparameter, and introduce the convergence properties of principal functions.

References

  • term1 Naimark, MA: Linear Differential Operators, II. Ungar, New York (1968)
  • term2 Lyance,VE: A differential operators with spectral singularities. I, II, AMS Translations 2(60), 185-225, 227-283 (1967)
  • term3 Keldysh, MV: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Soviet Mathematics-Doklady 77, 11-14 (1951)
  • term4 Keldysh, MV: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Russian Mathematical Surveys 26, 15-41 (1971)
  • term5 Aygar, Y, Bairamov, E: Jost solution and the spectral properties of the matrix-valued difference operators. Appl. Math. and Comput. 218, 9676-9681 (2012)
  • term6 Bairamov, E, Tunca, GB: Discrete spectrum and principal functions of non-selfadjoint differential operators. Journal of Computational and Applied Mathematics 235, 4519-4523 (2011)
  • term7 Adıvar, M, Bairamov, E: Spectral singularities of the nonhomogeneous Sturm-Liouville equations. Appl. Math. Lett. 15, 825-832 (2002)
  • term8 Bairamov, E, Cakar, O, Celebi, AO: Quadratic pencil of Schrödinger operators with spectral singularities: Discre spectrum and principal functions. J. Math. Anal. Appl. 216, 303-320 (1997)
  • term9 Krall, AM, Bairamov, E, Cakar, O: Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary conditions. J. Differ. Equ. 151, 252-267 (1999)
  • term10 Bairamov, E, Cakar, O, Krall, AM: An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. J. Differ. Equ. 151, 268-289 (1999)
  • Bairamov, E, Celebi, AO: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac Operators. Q. J. Math. 50, 371-384 (1999)
  • term11 Kır Arpat, E: An eigenfunctions expansions of the non-selfadjoint Sturm-Liouville operator with a singular potential. J. Math. Chemsitry 51, 2196-2213 (2013)
  • term12 Bairamov, E, Karaman, O: Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition. Acta Math. Hungar. 97, 121-131 (2002)
  • term13 Bairamov, E, Yokus, N: Spectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Abstr. Appl. Anal. 2009, Article ID 28959 (2009)
  • term14 Yokus, N, Koprubasi, T: Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter. J. Inequa. Appl. 2015, 1-7 (2015)
Year 2017, Volume: 30 Issue: 1, 371 - 379, 14.03.2017

Abstract

References

  • term1 Naimark, MA: Linear Differential Operators, II. Ungar, New York (1968)
  • term2 Lyance,VE: A differential operators with spectral singularities. I, II, AMS Translations 2(60), 185-225, 227-283 (1967)
  • term3 Keldysh, MV: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Soviet Mathematics-Doklady 77, 11-14 (1951)
  • term4 Keldysh, MV: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Russian Mathematical Surveys 26, 15-41 (1971)
  • term5 Aygar, Y, Bairamov, E: Jost solution and the spectral properties of the matrix-valued difference operators. Appl. Math. and Comput. 218, 9676-9681 (2012)
  • term6 Bairamov, E, Tunca, GB: Discrete spectrum and principal functions of non-selfadjoint differential operators. Journal of Computational and Applied Mathematics 235, 4519-4523 (2011)
  • term7 Adıvar, M, Bairamov, E: Spectral singularities of the nonhomogeneous Sturm-Liouville equations. Appl. Math. Lett. 15, 825-832 (2002)
  • term8 Bairamov, E, Cakar, O, Celebi, AO: Quadratic pencil of Schrödinger operators with spectral singularities: Discre spectrum and principal functions. J. Math. Anal. Appl. 216, 303-320 (1997)
  • term9 Krall, AM, Bairamov, E, Cakar, O: Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary conditions. J. Differ. Equ. 151, 252-267 (1999)
  • term10 Bairamov, E, Cakar, O, Krall, AM: An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. J. Differ. Equ. 151, 268-289 (1999)
  • Bairamov, E, Celebi, AO: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac Operators. Q. J. Math. 50, 371-384 (1999)
  • term11 Kır Arpat, E: An eigenfunctions expansions of the non-selfadjoint Sturm-Liouville operator with a singular potential. J. Math. Chemsitry 51, 2196-2213 (2013)
  • term12 Bairamov, E, Karaman, O: Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition. Acta Math. Hungar. 97, 121-131 (2002)
  • term13 Bairamov, E, Yokus, N: Spectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Abstr. Appl. Anal. 2009, Article ID 28959 (2009)
  • term14 Yokus, N, Koprubasi, T: Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter. J. Inequa. Appl. 2015, 1-7 (2015)
There are 15 citations in total.

Details

Journal Section Mathematics
Authors

Nihal Yokuş

Turhan Köprübaşı

Publication Date March 14, 2017
Published in Issue Year 2017 Volume: 30 Issue: 1

Cite

APA Yokuş, N., & Köprübaşı, T. (2017). PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION. Gazi University Journal of Science, 30(1), 371-379.
AMA Yokuş N, Köprübaşı T. PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION. Gazi University Journal of Science. March 2017;30(1):371-379.
Chicago Yokuş, Nihal, and Turhan Köprübaşı. “PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION”. Gazi University Journal of Science 30, no. 1 (March 2017): 371-79.
EndNote Yokuş N, Köprübaşı T (March 1, 2017) PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION. Gazi University Journal of Science 30 1 371–379.
IEEE N. Yokuş and T. Köprübaşı, “PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION”, Gazi University Journal of Science, vol. 30, no. 1, pp. 371–379, 2017.
ISNAD Yokuş, Nihal - Köprübaşı, Turhan. “PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION”. Gazi University Journal of Science 30/1 (March 2017), 371-379.
JAMA Yokuş N, Köprübaşı T. PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION. Gazi University Journal of Science. 2017;30:371–379.
MLA Yokuş, Nihal and Turhan Köprübaşı. “PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION”. Gazi University Journal of Science, vol. 30, no. 1, 2017, pp. 371-9.
Vancouver Yokuş N, Köprübaşı T. PRINCIPAL FUNCTIONS OF BOUNDARY-VALUE PROBLEM WITH QUADRATIC SPECTRAL PARAMETER IN BOUNDARY CONDITION. Gazi University Journal of Science. 2017;30(1):371-9.