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This paper addresses two different but related questions regarding anunbounded symmetric tridiagonal operator: its self-adjointness and theapproximation of its spectrum by the eigenvalues of its finite truncations. Thesufficient conditions given in both cases improve and generalize previouslyknown results. It turns out that, not only self-adjointness helps to studylimit points of eigenvalues of truncated operators, but the analysis of suchlimit points is a key help to prove self-adjointness. Several examples show theadvantages of these new results compared with previous ones. Besides, anapplication to the theory of continued fractions is pointed out.

Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations