The cubic eigenparameter dependent discrete Dirac equations with principal functions

Abstract

Let us consider the Boundary Value Problem (BVP) for the discrete Dirac Equations

{<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>

(γ₀+γ₁λ+γ₂λ²+γ₃λ³)y₁⁽²⁾+(β₀+β₁λ+β₂λ²+β₃λ³)y₀⁽¹⁾=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 and λ is a eigenparameter. Discussing the eigenvalues and the spectral singularities, we prove that the BVP (0.1), (0.2) has a finite number of eigenvalues and spectral singularities with a finite multiplicities, if

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

holds, for some ε>0 and (1/2)≤δ≤1.

Keywords

• Discrete Dirac equations,Eigenparameter,Spectral analysis,Spectrum,Principal functions

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