The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials
The authors consider the spectrum of the self-adjoint one-dimensional Schrödinger operator \(H = -d^2/dx^2 + V(x)\), acting on the Hilbert space \(L_2(0, \infty)\), with a boundary condition at \(x=0\). Considering the potential \(V\) as a perturbation, the authors present three theorems providing the conditions when suitable smallness assumptions on \(V\) guarantee stability of the absolutely continuous spectrum of \(H_0 = -d^2/dx^2\). In particular, the authors present a sharper version of a theorem of \textit{A. Kiselev} [Commun. Math. Phys. 179, No. 2, 377-400 (1996; Zbl 0858.34071)], stating that the absolutely continuous spectrum is preserved if \(V(x) = O(x^{-\alpha})\) \((\alpha >2/3)\).
scientific article; zbMATH DE number 1117493