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.