  We give an explicit formula for the wave operators for Schroedinger operators
on the half-line with a potential decaying strictly faster than the polynomial
of degree minus two. The formula consists of the main term given by the
scattering operator and a function of the generator of the dilation group, and
a Hilbert-Schmidt remainder term. Our method is based on the elementary
construction of the generalized Fourier transform in terms of the solutions of
the Volterra integral equations. As a corollary, a topological interpretation
of Levinson's theorem is established via an index theorem approach.
