= Solution
Complete the square to obtain, initially for real $y,z$,
$$
\int_{\mathbb R}e^{-t^2}G(t,y)G(t,z)dt=e^{-y^2-z^2}\int_{\mathbb R}e^{-t^2+2t(y+z)}dt=\sqrt\pi e^{2yz}.
$$
Compare coefficients of $y^mz^n$ on the two sides. The left coefficient is $(m!n!)^{-1}\int e^{-t^2}H_m(t)H_n(t)dt$, and the right is zero unless $m=n$, in which case it is $\sqrt\pi2^n/n!$. The polynomials are real on the real line, so these are the complex $L^2$ <inner products> of the $\phi_n$. Thus
$$
\boxed{\langle\phi_m,\phi_n\rangle=\delta_{mn}\sqrt\pi\,2^nn!,\qquad\|\phi_n\|_2=\pi^{1/4}\sqrt{2^nn!}.}
$$
Polynomial times Gaussian decay also directly shows each $\phi_n$ lies in $L^2$.
\Image[/past-exam-of-the-analysis-course-of-the-university-of-oxford/2007-hermite-functions.png]
{title=Normalized <Hermite functions> for orders zero through three}
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