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, $\gamma$, 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.