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 ].