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Zbl 0945.47026
Bairamov, Elgiz; Çelebi, A.Okay
Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators.
(English)
[J] Q. J. Math., Oxf. II. Ser. 50, No.200, 371-384 (1999). ISSN 0033-5606; ISSN 1464-3847/e

Let $L$ denote the non-self-adjoint discrete Dirac operator generator in $\ell_2(\bbfN,\bbfC^2)$ by \align y^{(2)}_{n+1}- y^{(2)}_n+ p_n y^{(1)}_n & = \lambda y^{(1)}_n,\\ - y^{(1)}_n+ y^{(1)}_{n-1}+ q_n y^{(2)}_n & = \lambda y^{(2)}_n,\quad n= 1,2,\dots\endalign and $$y^{(1)}_0= 0,$$ where the $p_n$, $q_n\in\bbfC$, $n= 1,2,\dots$\ . It is proved that if, for some $\varepsilon> 0$, $$\sup_{1\le n<\infty} (|p_n|+|q_n|)\exp(\varepsilon \sqrt n)< \infty.\tag 1$$ $L$ has a finite number of eigenvalues and spectral singularities (poles of the resolvent kernel which are not eigenvalues), each of finite multiplicity, and continuous spectrum $[-2,2]$. Under the condition (1) an integral representation is obtained for the Weyl (or, more accurately, the Hellinger-Nevanlinna) function, and this yields an expansion theorem in terms of the principal vectors of $L$.
[W.D.Evans (Cardiff)]
MSC 2000:
*47B39 Difference operators (operator theory)
47A10 Spectrum and resolvent of linear operators
47B25 Symmetric and selfadjoint operators (unbounded)

Keywords: spectrum; spectral expansion; Weyl function; non-self-adjoint discrete Dirac operator; eigenvalues; spectral singularities; principal vectors

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