Unique properties of Gaussians

[mnb96]

Hello,
let's suppose we are given a Gaussian function $$f(x)=Ae^{-ax^2}$$ (where a,A are real scalars and a is positive)

Is it possible to (dis)prove that the following identity is true only for Gaussians?

$$\frac{\partial^2}{\partial x^2} log f(x) = const$$

Thanks!

 

[betel]

If you solve the differential equation you end up with
$$f(x) = A e^{const\. x^2 + B x}.$$
So provided the constant is negative, you get a gaussian with a shift in the variable.