Sum rules for Jacobi matrices and their applications to spectral theory

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J -J 0 is Hilbert-Schmidt, and a proof of Nevai's conjecture that the Szeg condition holds if J -J 0 is trace class.