### 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,

| (32) |

where is a normalized gaussian. Introducing the Fourier-transforms and of
and , Eq. (32) takes the form

| (33) |

Non-trivial solutions for [and ] 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 with a leading quadratic term. Thus, the Fourier transform of every solution
of (32) must obey

| (34) |

for all . This implies that the Fourier-transform of , which is proportional to the second
derivative of , vanishes too. This implies , in agreement with
a theorem by Titchmarsh [24]. [The solution for is a linear combination of and
].