Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices (Q732091)

The present paper concerns a special class of symmetric Jacobi matrices \(J(\{a_n\},\{b_n\})\) and corresponding operators in \(\ell^2(\mathbb N)\). Specifically, let the diagonal entries \(b_n\) satisfy \(b_{n+1}-b_n\geq cn^\gamma\), off-diagonal entries \(a_n\) obey \(|a_n|\leq Cn^\beta\) with \(\gamma>\beta\geq 0\). By Janas and Naboko's criterion, the operator \(J\) has a compact resolvent, so its spectrum is discrete. The main result provides some asymptotics formulae for its eigenvalues by using the method of diagonalization and the notion of near-similarity in the sense of Rozenbljum. The examples in the last section, which appear in applications, illustrate nicely the main results of the paper.