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

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.

Keywords

• Eigenvalue,Jost solution,principal function,non-selfadjoint differential operator,spectral analysis,spectral singularity.

References

1. term1 Naimark, MA: Linear Differential Operators, II. Ungar, New York (1968)
2. term2 Lyance,VE: A differential operators with spectral singularities. I, II, AMS Translations 2(60), 185-225, 227-283 (1967)
3. term3 Keldysh, MV: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Soviet Mathematics-Doklady 77, 11-14 (1951)
4. term4 Keldysh, MV: On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Russian Mathematical Surveys 26, 15-41 (1971)
5. 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)
6. 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)
7. term7 Adıvar, M, Bairamov, E: Spectral singularities of the nonhomogeneous Sturm-Liouville equations. Appl. Math. Lett. 15, 825-832 (2002)
8. 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)

9. 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)
10. 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)
11. Bairamov, E, Celebi, AO: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac Operators. Q. J. Math. 50, 371-384 (1999)
12. 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)
13. 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)
14. term13 Bairamov, E, Yokus, N: Spectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Abstr. Appl. Anal. 2009, Article ID 28959 (2009)
15. 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)