Eigenfunction

[Poirot]

show that \(\displaystyle e^{-0.5x^2}\) is an eigenfunction of the operator \(\displaystyle \frac{d^2}{dx}-x^2\) and finds it's eigenvalue. I get \(\displaystyle e^{-0.5x^2}(x^2-1)-x^2\) so it doesn't seem like its an eigenfunction.

 

[Evgeny.Makarov]

Re: eigenfunction

The operator is defined as follows.

\[
\left(\frac{d^2}{dx^2}-x^2\right)f(x)=\frac{d^2f(x)}{dx^2}-x^2f(x).
\]