Spectral properties of the second order difference equation with selfadjoint operator coefficients

Abstract

In this paper, we consider the second order difference equation defined on the whole axis with selfadjoint operator coefficients. The main objective of this study is to obtain the continuous and discrete spectrum of the discrete operator which is generated by this difference equation. To achieve this, we first obtain the Jost solutions of this equation explicitly and then examine the analytical and asymptotic properties of these solutions. With the help of these properties we find the continuous and discrete spectrum of this operator. Finally we obtain the sufficient condition which ensures that this operator has a finite number of eigenvalues.

Keywords

• Difference equations,Jost solution,operator coefficients,continuous spectrum

References

1. Adivar, M. and Bairamov, E., Spectral properties of non-selfadjoint difference operators, J. Math. Anal. Appl., 261, (2001), 461-478.
2. Agarwal, R. P., Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics, New York, Marcel Dekker, 2000.
3. Agarwal, R. P. and Wong, P. J. Y. Advanced Topics in Difference Equations, Mathematics and Its Applications, Dordrecht: Kluwer Academic Publishers Group, 1997.
4. Aygar, Y. and Bairamov, E., Jost solution and the spectral properties of the matrix-valued difference operators, Appl. Math. Comput., 218, (2012), 9676-9681.
5. Bairamov, E., Aygar, Y. and Cebesoy, S., Spectral analysis of a selfadjoint matrix-valued discrete operator on the whole axis, J. Nonlinear Sci. Appl., 9, (2016), 4257-4262.
6. Carlson, R., An inverse problem for the matrix Schrödinger equation, J. Math. Anal. Appl., 267, (2002), 564-575.
7. Cebesoy, S., Aygar, Y. and Bairamov, E., Matrix-valued difference equations with spectral singularities, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 9 (11), (2015), 658-661.
8. Clark, S., Gesztesy, F. and Renger, W., Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators, J. Diferential Equations, 219, (2005), 144-182.

9. Gasymov, M. G., Zikov, V. V. and Levitan, B. M., Conditions for discreteness and finiteness of the negative spectrum of Schrödinger's operator equation, Mat. Zametki., 2, (1967), 531-538 (in Russian).
10. Gesztesy, F., Kiselev, A. and Makarov, K. A., Uniqueness results for matrix-valued Schrodinger, Jacobi and Dirac-type operators, Math. Nachr., 239, (2002), 103-145.
11. Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations, Jerusalem, 1965.
12. Kostjucenko, A. G. and Levitan, B. M., Asymptotic behavior of eigenvalues of the operator Sturm-Liouville problem, Funkcional. Anal. i Prilozen., 1, (1967), 86-96 (in Russian).
13. Levitan, B. M., Investigation of the Green's function of a Sturm-Liouville equation with an operator coefficient, Mat. Sb. (N.S.), 76 (118), (1968), 239-270 (in Russian).
14. Levitan, B. M. and Sargsyan, I. S., Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, American Mathematical Society (Translated from Russian), 1975.
15. Levitan, B. M. and Suvorcenkova, G. A., Sufficient conditions for discreteness of the spectrum of a Sturm-Liouville equation with operator coefficient, Funkcional. Anal. i Prilozen, 2 (2), 1968, 56-62 (in Russian).
16. Lusternik, L. A. and Sobolev, V. I., Elements of Functional Analysis, Halsted Press, New York, 1974.
17. Marchenko V. A., Sturm-Liouville Operators and Applications, Birkhauser Verlag, Basel, 1986.
18. Naimark M. A., Linear Differential Operators, I-II, Ungar, New York, 1968.
19. Serebrjakov, V. P., An inverse problem of the scattering theory for difference equations with matrix coefficients, Dokl. Akad. Nauk SSSR., 250 (3), 1980, 562-565 (in Russian).