Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $LH_n (x) = -nH_n (x)$.
The only proof of this miracle that I know of goes through solving the SL problem by power-series method, and on the other hand obtaining the orthogonal polynomials via Gram-Schmidt algorithm.
My question: I am going to give a talk about this property for graduate students. Is there another, perhaps a more "natural" explanation for this miracle?
(this is related to a previous question of mine, but not entirely the same)