Wave operators and positive eigenvalues for a Schrödinger equation with oscillating potential

A study is made of the Schrödinger equation on a half,axis with a potential $q(x)$ that is not absolutely integrable and may be unbounded at infinity. The main result of the paper is the proof of the existence and completeness of the wave operators $W_{\pm}(H,H_0)$ under the condition that the Fourier transform of the potential at the upper limit converges sufficiently fast everywhere except at a certain discrete set of points $k_j$. It is also proved that for such potentials eigenvalues in the continuous spectrum can appear only at the points $\lambda_j=k_j^2/4$.