10 Appendix

Using the theory of generalized functions [15] the integral equation (30) can be solved quickly. We discuss for simplicity the one-dimensional integral equation,
            ∫

f (x  ) =        dxk   (x   -  y )f  (y  ),
(32)

where k (x ) is a normalized gaussian. Introducing the Fourier-transforms F  (k ) and K   (k ) of f  (x ) and k (x  ) , Eq. (32) takes the form
∫
                 [       √   -----       ]

    dkF    (k  )   1  -      2 πK    (k )   =   0.
(33)

Non-trivial solutions for F  (k ) [and f (x  ) ] can only exist if the term in brackets has zeros. This is the case since the Fourier-transform of a normalized gaussian is again a normalized gaussian and therefore the bracket vanishes at k  =   0 with a leading quadratic term. Thus, the Fourier transform of every solution of (32) must obey
F  (k )k2   =   0.
(34)

for all k . This implies that the Fourier-transform of   2
k   F  (k ) , which is proportional to the second derivative   ′′
f   (x  ) of f  (x ) , vanishes too. This implies f (x  ) =   a  +   bx , in agreement with a theorem by Titchmarsh [24]. [The solution for F  (k  ) is a linear combination of δ (k ) and   ′
δ  (k  ) ].