Gökhan MUTLU

Spectrum of Discrete Sturm-Liouville Equation with Self-adjoint Operator Coefficients on the Half-line

We investigate the spectrum of the Sturm-Liouville difference equation on the half-line with self-adjoint operator coefficients in an infinite dimensional Hilbert space together with the Dirichlet boundary condition. We find the Jost solution and examine its analytical and asymptotical properties. Using these properties, we obtain the continuous and point spectrum of the discrete operator generated by the Sturm-Liouville difference equation with self-adjoint operator coefficients. We also show that this operator has a finite number of eigenvalues with finite multiplicities under a certain condition on the operator coefficients.

Keywords:

Discrete operator, eigenvalues, selfadjoint operators Sturm-Liouville difference equation,

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Aktosun T, Weder R, 2020. Direct and Inverse Scattering for the Matrix Schrödinger Equation, Applied Mathematical Sciences, 203, Springer, Cham.
Aygar Y, Bairamov E, 2012. Jost solution and the spectral properties of the matrix-valued difference operators. Applied Mathematics and Computation, 218: 9676-9681.
Bairamov E, Aygar Y, Cebesoy S, 2016. Spectral analysis of a selfadjoint matrix-valued discrete operator on the whole axis. Journal of Nonlinear Sciences and Applications, 9: 4257-4262.
Bairamov E, Kir Arpat E, Mutlu G, 2017. Spectral properties of non-selfadjoint Sturm-Liouville operator with operator coefficient. Journal of Mathematical Analysis and Applications, 456: 293-306.
Gasymov MG, Zikov VV, Levitan BM, 1967. Conditions for discreteness and finiteness of the negative spectrum of Schrödinger's operator equation. Matematicheskie Zametki, 2: 531-538.
Glazman IM, 1965. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations, Jerusalem.
Keldysh MV, 1971. On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Russian Mathematical Surveys, 26: 15-44.
Kir Arpat E, Mutlu G, 2015. Spectral properties of Sturm-Liouville system with eigenvalue-dependent boundary conditions. International Journal of Mathematics, 26: 1550080-1550088.
Lusternik LA, Sobolev VI, 1974. Elements of Functional Analysis. Halsted Press, New York.
Mutlu G, 2020. Spectral properties of the second order difference equation with selfadjoint operator coefficients. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 69: 88-96.
Mutlu G, Kir Arpat E, 2020a. Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis. Hacettepe Journal of Mathematics and Statistics, 49: 1686-1694.
Mutlu G, Kir Arpat E, 2020b. Spectral analysis of non-selfadjoint second order difference equation with operator coefficient. Sakarya University Journal of Science, 24: 494-500.
Serebrjakov VP, 1980. An inverse problem of the scattering theory for difference equations with matrix coefficients. Doklady Akademii Nauk SSSR, 250(3): 562-565 (in Russian).
Yokus N, Coskun N, 2019. Spectral properties of discrete Klein-Gordon s-wave equation with quadratic eigenparameter-dependent boundary condition. Iranian Journal of Science and Technology, Transactions A: Science, 43: 1951-1955.
Yokus N, Coskun N, 2020. Spectral analysis of Klein-Gordon difference operator given by a general boundary condition. Communications in Mathematics and Applications, 11(2): 271-279.