  In the presence of a magnetic impurity the spin-up and down band states are
modified differently by the impurity. If the multi-electron scalar product
(MESP) between the occupied spin-up and down states approaches zero then this
defines an orthogonality catastrophe. In the present paper the MESP is
investigated for the FAIR (Friedel Artificial Iserted Resonance) solution for a
Friedel-Anderson impurity. A basis of Wilson states is used. The MESP is
numerically determined for the (enforced) magnetic, the singlet, and the
triplet states as a function of the number N of Wilson states. The magnetic and
the triplet state show an exponentially decreasing MESP as a function of N.
Surprisingly it is not the number of states which causes this decrease. It is
instead the energy separation of the highest occupied state from the Fermi
energy which determines the reduction of the MESP. In the singlet state the
ground-state requires a finite MESP to optimize its energy. As a consequence
there is no orthogonality catastrophe. The MESP approaches a saturation value
as function of N.
