Difference equations of second order with spectral singularities
Spectral properties of the discrete eigenvalue problem \[ a_{n}y_{n+1}+b_ny_n+a_{n-1}y_{n-1}=\lambda y_n,\quad n\in \mathbb Z, \tag{*} \] where \(a,b\) are complex sequences with \(a_n\neq 0\) satisfying \(\sum_{n\in \mathbb Z}|n|(|1-a_n|+|b_n|)<\infty \), and \(\lambda\) is the spectral parameter, are investigated. Eigenvalues and spectral singularities of (*) are defined via zeros of the Wronskian of solutions of this equation. The main result of the paper reads as follows. Theorem: Suppose that for some \(\varepsilon>0\) and \(1/2\leq \delta<1\) \[ \sup_{n\in \mathbb Z}\left\{e^{\varepsilon|n|^{\delta}}\left(|1-a_n|+|b_n|\right)\right\}<\infty. \] Then (*) has only a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity. A related paper of the same authors is published in [J. Math. Anal. Appl. 261, 461-478 (2001; Zbl 0992.39018)], where the statement of Theorem is proved under a stronger assumption.
scientific article; zbMATH DE number 1901866