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We study half-line Schrödinger operators with locally H^{-1} potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last-Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the second part, we focus on potentials which are decaying in a local H^{-1} sense; we establish a spectral transition between short-range and long-range potentials and an ell^2 spectral transition for sparse singular potentials. The regularization procedure used to handle distributional potentials is also well suited for controlling rapid oscillations in the potential; thus, even within the class of smooth potentials, our results apply in situations which would not classically be considered decaying or even bounded.
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There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.more » « less
