Generalized Fourier Transform for Schrِdinger Operators with Potentials of Order Zero

We investigate the Schrödinger operator H=−Δ+V acting in L2(Rn), n⩾2, for potentials V that satisfy ∂αxV(x)=O(|x|−|α|) as |x|→∞. By introducing coordinates on Rn closely related to a relevant eikonal equation we obtain an eigenfunction expansion for H at high energies.