I am searching for tips towards a proof of the completeness property for the solutions of the harmonic oscillator based on real Hermite polynomials $H_n(x)$, i.e.:
$$e^{-(x^2+a^2)/2}\sum_{n}^\infty \frac{H_n(x)H_n(a)}{2^n n!\sqrt{\pi}}=\delta(x-a)$$
Screening the literature I often end up at the Christoffel–Darboux formula, but I am unable to see a relation of the rhs towards the delta-distribution. Do I need an expansion/approximation of $\delta(x)$?
I am lost and thankful for any tip, hint, literature.