Spectral properties of some Jacobi matrices with double weights.
Jacobi matrices with zero main diagonal are considered: these are linear operators on \(\ell^2(\mathbb{N})\) on the maximal domain of definition whose matrix \([J_{m,n}]^\infty_{m,n=1}\) with respect to the standard basis of \(\ell^2(\mathbb{N})\) is bidiagonal and conjugate transpose: \(J_{m,m+1}= \lambda_m= \overline{J_{m+1,m}}\), and \(J_{m,n}= 0\) if \(| m-n|\neq 1\). It is assumed that the \(\lambda_m\)'s are real and nonzero. Two special classes of Jacobi matrices with zero main diagonal and double weights are studied: For a positive sequence \(\alpha= \{\alpha_n\}^\infty_{n=1}\) and \(\varepsilon> 0\), the Jacobi matrices with zero main diagonal \(J_\alpha\) and \(J_{\varepsilon,\alpha}\) are defined by \(\lambda_{2n-1}= \lambda_{2n}= \alpha_n\) and by \(\lambda_1= \varepsilon\), \(\lambda_{2n}= \lambda_{2n+1}= \alpha_n\), respectively. It is known that for any choice of \(\alpha\) and \(\varepsilon\) the matrices \(J_\alpha\) and \(J_{\varepsilon,\alpha}\) are selfadjoint. The main results of the present paper concern absolute continuity of \(J_\alpha\) and \(J_{\varepsilon,\alpha}\) in the cases \(\alpha_n= n^a\) for a fixed \(a\), and absolute continuity of Jacobi matrices obtained by certain perturbations of these special cases. Applications are given to Jacobi matrices that appear in certain birth and death processes [\textit{J. Janas} and \textit{S. Naboko}, Oper. Theory, Adv. Appl. 127, 387--397 (2001; Zbl 0996.47040)]. It is proved that Jacobi matrices in the birth and death processes are selfadjoint and absolutely continuous, and their spectra are determined.
scientific article; zbMATH DE number 1917782