Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices

In this article we calculate the asymptotic behaviour of the point spectrum for some special self-adjoint unbounded Jacobi operators J   acting in the Hilbert space l²=l²(N)$l^2=l^2(\mathbb{N})$. For given sequences of positive numbers _λn${\lambda }_n$ and real _qn$q_n$ the Jacobi operator is given by J=SW+WS^*+Q$J=\text{<italic>SW</italic>}+{\mathit{WS}}^{*}+Q$, where Q=diag(_qn)$Q=\mathrm{diag}(q_n)$ and W=diag(_λn)$W=\mathrm{diag}({\lambda }_n)$ are diagonal operators, S is the shift operator and the operator J   acts on the maximal domain. We consider a few types of the sequences {_qn}$\{q_n\}$ and {_λn}$\{{\lambda }_n\}$ and present three different approaches to the problem of the asymptotics of eigenvalues of various classes of J's. In the first approach to asymptotic behaviour of eigenvalues we use a method called successive diagonalization, the second approach is based on analytical models that can be found for some special J's and the third method is based on an abstract theorem of Rozenbljum.