= Hermite functions are Fourier eigenfunctions
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{title2=$\mathcal F(H_n(x)e^{-x^2/2})=(-i)^nH_n(x)e^{-x^2/2}$}
Use the unitary <Fourier transform> with phase $e^{-i\lambda x}$. It commutes with $-d^2/dx^2+x^2$, because differentiation transforms to multiplication by $i\lambda$ and multiplication by $x$ transforms to $i\,d/d\lambda$. The <Hermite differential equation> has a unique polynomial solution of each degree up to scale, and multiplying by the Gaussian gives a <Hermite function>. Transforming a polynomial times this Gaussian again gives a polynomial times the Gaussian of the same degree; comparison of leading coefficients gives the eigenvalue $(-i)^n$.
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