Principal functions of nonselfadjoint discrete Dirac equations with spectral parameter in boundary conditions. (English) Zbl 1253.39005

Aygar, Yelda; Olgun, Murat; Koprubasi, Turhan

Abstr. Appl. Anal. 2012, Article ID 924628, 15 p. (2012).

Summary: Let \(L\) denote the operator generated in \(\ell_2(\mathbb N, \mathbb C^2)\) by \(a_{n+1}y^{(2)}_{n+1} + b_ny^{(2)}_n + p_ny^{(1)}_n = \lambda y^{(1)}_n, a_{n-1}y^{(1)}_{n-1} + b_ny^{(1)}_n + q_ny^{(2)}_n = \lambda y^{(2)}_n, n \in \mathbb N\), and the boundary condition \((\gamma_0 + \gamma_1 \lambda)y^{(2)}_1 + (\beta_0 + \beta_1 \lambda)y^{(1)}_0 = \mathbf{0}\), where \((a_n), (b_n), (p_n)\) and \((q_n), n \in \mathbb N\) are complex sequences, \(\gamma_i, \beta_i \in \mathbb C, i = 0, 1\), and \(\lambda\) is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of \(L\).

MSC:

39A12

Discrete version of topics in analysis

Keywords:

nonselfadjoint discrete Dirac equations

References:

[1]	M. A. Naĭmark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis,” American Mathematical Society Translations, vol. 16, pp. 103-193, 1960. · Zbl 0100.29504
[2]	V. E. Lyance, “A differential operator with spectral singularities, I, II,” American Mathematical Society Translations, vol. 2, no. 60, pp. 185-225, 1967. · Zbl 0189.38102
[3]	M. Adıvar, “Quadratic pencil of difference equations: jost solutions, spectrum, and principal vectors,” Quaestiones Mathematicae, vol. 33, no. 3, pp. 305-323, 2010. · Zbl 1274.39039
[4]	M. Adıvar and M. Bohner, “Spectral analysis of q-difference equations with spectral singularities,” Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 695-703, 2006. · Zbl 1134.39018
[5]	M. Adivar and M. Bohner, “Spectrum and principal vectors of second order q-difference equations,” Indian Journal of Mathematics, vol. 48, no. 1, pp. 17-33, 2006. · Zbl 1114.39003
[6]	E. Bairamov and A. O. \cCelebi, “Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,” The Quarterly Journal of Mathematics, vol. 50, no. 200, pp. 371-384, 1999. · Zbl 0945.47026
[7]	E. Bairamov and C. Coskun, “Jost solutions and the spectrum of the system of difference equations,” Applied Mathematics Letters, vol. 17, no. 9, pp. 1039-1045, 2004. · Zbl 1072.39013
[8]	E. Bairamov, Y. Aygar, and T. Koprubasi, “The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4519-4523, 2011. · Zbl 1229.39006
[9]	M. Adıvar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 461-478, 2001. · Zbl 0992.39018
[10]	E. Bairamov and T. Koprubasi, “Eigenparameter dependent discrete Dirac equations with spectral singularities,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4216-4220, 2010. · Zbl 1191.39005