MHB Eigenfunction of \frac{d^2}{dx}-x^2: e^{-0.5x^2} with eigenvalue x^2-1

[Poirot1]

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

 

[Evgeny.Makarov]

Re: eigenfunction

The operator is defined as follows.

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