  We derive rigorously the leading asymptotics of the so-called Anderson
integral in the thermodynamic limit for one-dimensional, non-relativistic,
spin-less Fermi systems. The coefficient, $\gamma$, of the leading term is
computed in terms of the S-matrix. This implies a lower and an upper bound on
the exponent in Anderson's orthogonality catastrophe, $\tilde
CN^{-\tilde\gamma}\leq \mathcal{D}_N\leq CN^{-\gamma}$ pertaining to the
overlap, $\mathcal{D}_N$, of ground states of non-interacting fermions.
