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

We prove that for any one-dimensional Schrödinger operator with potential V(x) satisfying decay condition |V(x)| ≦ Cx-3/4-ε, the absolutely continuous spectrum fills the whole positive semi-axis. The description of the set in ℝ+ on which the singular part of the spectral measure might be supported is also given. Analogous results hold for Jacobi matrices.

Duke Scholars

Author Alexander A. Kiselev Mathematics