Infinite Jacobi matrices with unbounded entries: Asymptotics of eigenvalues and the transformation operator approach

In this paper the exact asymptotics of eigenvalues lambda(n)(J), n-->infinity, of a class of unbounded self-adjoint Jacobi matrices J with discrete spectrum are given. Their calculation is based on a successive diagonalization approach-a new version of the classical transformation operator method. The approximations of the transformation operator are constructed step by step using a successive diagonalization procedure, which results in higher order approximations of the lambda(n)(J).