= Gaussian polynomial operator
{c}
{title2=$Tf=f''-2xf'-2f$}
On $\mathbb C[x]$, equip <polynomials> with the <inner product>
$$
\langle f,g\rangle=\int_{-\infty}^{\infty}\overline{f(x)}g(x)e^{-x^2}\,dx.
$$
The <linear operator>
$$
Tf=\frac{d}{dx}\left(e^{x^2}\frac{d}{dx}(e^{-x^2}f)\right)
=f''-2xf'-2f
$$
preserves the space $V_n$ of <polynomials> of <polynomial degree> at most $n$. <Integration by parts>, whose boundary terms vanish by Gaussian decay, gives
$$
\langle Tf,g\rangle=-\langle f',g'\rangle-2\langle f,g\rangle
=\langle f,Tg\rangle.
$$
Thus $T$ equals its <adjoint on an inner product space>. If a nonzero <polynomial> of degree $n$ is an <eigenvector>, its leading coefficient forces its <eigenvalue> to be $-2(n+1)$.
One can construct all these <eigenvectors> without solving a <differential equation>. The one-dimensional space $L_n=V_n\cap V_{n-1}^{\perp}$ is invariant: for $u\in V_{n-1}$ and $v\in L_n$, the displayed identity gives $\langle u,Tv\rangle=\langle Tu,v\rangle=0$. Its unit vector of positive real leading coefficient is therefore an <eigenvector> of degree $n$. These choices are exactly the <Gram-Schmidt process> applied to $1,x,x^2,\ldots$, producing normalized <Hermite polynomials>. Their first three members are
$$
g_0=\pi^{-1/4},\qquad g_1=\sqrt2\pi^{-1/4}x,\qquad
g_2=\frac{2x^2-1}{\sqrt2\pi^{1/4}}.
$$
Every <polynomial> is a finite linear combination of these orthonormal <eigenvectors>.
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