## Background

Working on a quantum mechanics problem, I've stumbled on the problem of maximizing the functional $$\int_{A} \varphi_m \varphi_n$$ in the limit of large $m$ and $n$, given that $n \gg m$. The function $\varphi_m$ is the $m$-th Hermite function, defined as $$\varphi_m(x) = (2^m m! \sqrt{\pi})^{-1/2} e^{-x^2/2} H_m(x),$$ so that $\|\varphi_m\|_2 = 1$.

It's clear that this problem is equivalent to evaluating the function $$ f(m,n) = \int_{\mathbb{R}} |\varphi_m \varphi_{n}|,$$ which appears to me to be simpler, so I've concentrated myself on it. It's clear that it's not sensible to search for an exact formula, as the solution gets very complicated very quickly, so I'm only looking for an asymptotic expansion.

Only knowing if it goes to zero in this limit would be also very interesting; I have physical reason to think it does.

Using the asymptotic form of the Hermite functions I managed to prove that $$f(m,n) \sim \frac{2}{\pi} \sqrt[4]{\frac{2}{n\pi^2}}\int_{\mathbb{R}}|\varphi_m|.$$ But an asymptotic form for the right-hand side eludes me, so

Is there an asymptotic form for $\|\varphi_m\|_1$?