Commun. Math. Phys. 193, 151 – 170 (1998) Communications in Mathematical Physics c  Springer-Verlag 1998 The Absolutely Continuous Spectrum of One-Dimensional Schr¨ odinger Operators with Decaying Potentials Christian Remling Universit¨ at Osnabr¨ uck, Fachbereich Mathematik/Informatik, 49069 Osnabr¨ uck, Germany. E-mail: cremling@mathematik.uni-osnabrueck.de Received: 25 June 1997 / Accepted: 29 July 1997 Abstract: We investigate one-dimensional Schr¨ odinger operators with asymptotically small potentials. It will follow from our results that if |V (x)|≤ C(1+x) −α with α> 1/2, then  ac = (0, ∞) is an essential support of the absolutely continuous part of the spectral measure. We also prove that if C := lim sup x→∞ x |V (x)| < ∞, then the spectrum is purely absolutely continuous on ((2C/π) 2 , ∞). These results are optimal. 1. Introduction In this paper, I am interested in one-dimensional Schr¨ odinger equations, − y ′′ (x)+ V (x)y(x)= Ey(x), (1) with asymptotically small potentials V (x). We will treat only the half-line problem x ∈ [0, ∞) explicitly (of course, the results below extend easily to whole-line problems). So, we are interested in the spectral properties of the operators H β = − d 2 dx 2 + V (x) on L 2 (0, ∞). The index β ∈ [0,π) refers to the boundary condition y(0) cos β +y ′ (0) sin β = 0. Although the emphasis will be on the continuous case, we will also occasionally discuss the discrete analogue of (1), y(n − 1) + y(n + 1) + V (n)y(n)= Ey(n). The properties of the corresponding classical system very naturally lead to the ques- tion of whether suitable smallness assumptions on V (x) at large x imply absence of singular spectrum on (0, ∞) or, at least, existence of absolutely continuous spectrum. Indeed, one can make the elementary remark that the spectrum is purely absolutely con- tinuous on (0, ∞) if V ∈ L 1 . On the other hand, the classical von Neumann-Wigner example [32] shows that potentials V (x)= O(1/x) can have embedded eigenvalues.