Limit points of eigenvalues of truncated unbounded tridiagonal operators

Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e _n } _n=1 ^∞ , σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T _N . We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.