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 Szegő condition holds if J − J 0 is trace class.

Abstract. We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation iut + ∆u = ±|u| 2 u for large spherically symmetric L 2 x (R 2 ) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.

Abstract. We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely α-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

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We establish global well-posedness and scattering for solutions to the mass-critical nonlinear SchrödingerIn the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.