Spectral Analysis Of Discrete Dirac Equation With Generalized Eigenparameter In Boundary Condition

Abstract

Let L denote the discrete Dirac operator generated in ℓ₂(ℕ,ℂ²) by the non-selfadjoint difference operators of first order

{<K1.1/>┊ #0.1

<K1.1 ilk="MATRIX" >
a_{n+1}y_{n+1}⁽²⁾+b_{n}y_{n}⁽²⁾+p_{n}y_{n}⁽¹⁾=λy_{n}⁽¹⁾
a_{n-1}y_{n-1}⁽¹⁾+b_{n}y_{n}⁽¹⁾+q_{n}y_{n}⁽²⁾=λy_{n}⁽²⁾ , n∈ℕ,
</K1.1>
with boundary condition

∑_{k=0}^{p}(y₁⁽²⁾γ_{k}+y₀⁽¹⁾β_{k})λ^{k}=0, #0.2

where (a_{n}), (b_{n}), (p_{n}) and (q_{n}), n∈ℕ are complex sequences, γ_{i}, β_{i}∈ℂ, i=0,1,2,...,p and λ is a eigenparameter. We discuss the spectral properties of L and we investigate the properties of the spectrum and the principal vectors corresponding to the spectral singularities of L, if

∑_{n=1}^{∞}exp(εn^{δ})(|1-a_{n}|+|1+b_{n}|+|p_{n}|+|q_{n}|)<∞

holds for some ε>0 and δ∈[(1/2),1].