Absolutely continuous spectrum of one-dimensional Schrodinger operators and Jacobi matrices with slowly decreasing potentials

Citation

Kiselev, Alexander A. (1997) Absolutely continuous spectrum of one-dimensional Schrodinger operators and Jacobi matrices with slowly decreasing potentials. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4XQY-8Q92. https://resolver.caltech.edu/CaltechETD:etd-09072005-112344

Abstract

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We show that for one-dimensional Schrodinger operators with potentials V(x) satisfying the decay condition [...], the absolutely continuous spectrum fills the whole positive semi-axis. We also give the description of a set of zero Lebesgue measure on which the embedded singular part of the spectral measure may be supported. Under additional conditions on the integrability of the potential, we show that potentials decaying as [...] also lead to the absolutely continuous spectrum of the Hamiltonian.

An analog of the short-range Jost functions is introduced for the square integrable potentials. The formula for the projection on the absolutely continuous component of the spectrum is derived for a certain class of power decaying potentials.

Some further applications of the introduced technique are given. We also show that similar results hold for Jacobi matrices.

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