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Eigenparameter dependent discrete Dirac equations with spectral singularities. (English) Zbl 1191.39005

The authors consider the discrete boundary value problem (BVP)

${a}_{n-1}{y}_{n-1}+{b}_{n}{y}_{n}+{a}_{n}{y}_{n+1}=\lambda {y}_{n},\phantom{\rule{1.em}{0ex}}n\in ℕ=\left\{1,2,\cdots \right\},\phantom{\rule{1.em}{0ex}}{y}_{0}=0,$

where $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ are complex sequences, ${a}_{0}\ne 0$ and $\lambda$ is a spectral parameter. They prove that the spectrum of the BVP consists of the continuous spectrum, the eigenvalues and the spectral singularities. They show that spectral singularities are poles of the resolvent and those are also embedded in the continuous spectrum, but indicating that they are not eigenvalues.

MSC:
 39A12 Discrete version of topics in analysis 81Q05 Closed and approximate solutions to quantum-mechanical equations 34L05 General spectral theory for OD operators