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Anderson's Orthogonality Catastrophe for One-dimensional Systems

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, $γ$, 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, $\tilde CN^{-\tildeγ}\leq \mathcal{D}_N\leq CN^{-γ}$ pertaining to the overlap, $\mathcal{D}_N$, of ground states of non-interacting fermions.