 Gaussian polynomial operator

On $\mathbb{C}[x]$, equip polynomials with the inner productThe linear operatorpreserves the space $V_n$ of polynomials of polynomial degree at most $n$. Integration by parts, whose boundary terms vanish by Gaussian decay, givesThus $T$ 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 $⟨u,Tv⟩=⟨Tu,v⟩=0$. 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 $1,x,x^2,\dots$, producing normalized Hermite polynomials. Their first three members areEvery polynomial is a finite linear combination of these orthonormal eigenvectors.