Background
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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 $\|\varphi\_m\|\_2 = 1$.




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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 $\|\varphi\_m\|\_1$?
> 
> 
> 



Asymptotic form of $L^1$-norm of Hermite functions
Results for the $L^p$ norms of general orthogonal polynomials have been given in Ref. [1]. 


In the case of the orthonormal Hermite polynomials $H\_n$, it was given in [1] for $L^{2p}$ norms, $0<p<4/3$, and then in [2], for any $0<p$. Let
$$N\_n(p)=\int\_{\mathbb R}|H\_n(x)e^{-x^2/2}|^{2p}dx.$$
Then, for $p<4/3$, as $n\to\infty$,
$$N\_n(p)=\left(\frac2\pi\right)^p\frac{\Gamma(p+1/2)}{\Gamma(p+1)}
\frac{\Gamma(1-p/2)}{\Gamma(3/2-p/2)}(2n)^{(1-p)/2}(1+o(1)).
$$



> 
> [1] A. I. Aptekarev, V. S. Buyarov, and J. S. Dehesa, Asymptotic
>  behavior of the Lp-norms and the entropy for general orthogonal
>  polynomials, Russian Acad. Sci. Sb. Math. 82 (1995), 373-395.
> 
> 
> [2] Aptekarev, A. I.; Dehesa, J. S.; Sánchez-Moreno, P.; Tulyakov, D.
>  N.; Asymptotics of Lp-norms of Hermite polynomials and Rényi entropy
>  of Rydberg oscillator states. Recent advances in orthogonal
>  polynomials, special functions, and their applications, 19–29, 
>  Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012.
> 
> 
> 


