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Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. (English) Zbl 0945.47026

Let L denote the non-self-adjoint discrete Dirac operator generator in 2 (, 2 ) by

y n+1 (2) -y n (2) +p n y n (1) =λy n (1) ,-y n (1) +y n-1 (1) +q n y n (2) =λy n (2) ,n=1,2,


y 0 (1) =0,

where the p n , q n , n=1,2, . It is proved that if, for some ε>0,

sup 1n< (|p n |+|q n |)exp(εn)<·(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.

47B39Difference operators (operator theory)
47A10Spectrum and resolvent of linear operators
47B25Symmetric and selfadjoint operators (unbounded)