Limit points of eigenvalues of truncated tridiagonal operators

Let T be the tridiagonal operator Ten=anen+1+an−1en−1+bnen, Te1=a1e2+b1e1, acting on a fixed orthonormal basis {en}, n=1,2,…, of a Hilbert space H. Let PN be the orthogonal projection on the finite-dimensional space HN spanned by the elements {e1,e2,…,eN} and let TN be the truncated operator TN=PNTPN. If T has a unique self-adjoint extension then the set Λ(T)={λ: there exists a sequence of eigenvalues λN of TN with the property λN→λ} contains the spectrum σ(T) of T and examples show that, in general, σ(T)≠Λ(T). For many reasons, the knowledge of the equality σ(T)=Λ(T) is important. In this paper sufficient conditions are presented such that σ(T)=Λ(T).