  We give an upper bound on the modulus of the ground-state overlap of two
non-interacting fermionic quantum systems with $N$ particles in a large but
finite volume $L^d$ of $d$-dimensional Euclidean space. The underlying
one-particle Hamiltonians of the two systems are standard Schr\"odinger
operators that differ by a non-negative compactly supported scalar potential.
In the thermodynamic limit, the bound exhibits an asymptotic power-law decay in
the system size $L$, showing that the ground-state overlap vanishes for
macroscopic systems. The decay exponent can be interpreted in terms of the
total scattering cross section averaged over all incident directions. The
result confirms and generalises P. W. Anderson's informal computation [Phys.
Rev. Lett. 18, 1049--1051 (1967)].
