##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 $\|\varphi_m\|_1$?

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 $\|\varphi_m\|_1$?

##Background##

Working on a quantum mechanics problem, I've stumbled on the problem of maximising the functional $$\int_A \varphi_m \varphi_n$$ in the limit of large $m$ and $n$, given that $n \gg m$. $\varphi_m$ 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 rhs eludes me, so

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

##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 $\|\varphi_m\|_1$?

I'm interest in an asymptotic expansion of the function $$ f(m,n) = \int_{\mathbb{R}} |\varphi_m \varphi_{n}| $$ in the limit of large $m$ and $n$, given that $n \gg m$. $\varphi_m$ 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 $\|\varphi_m\|_2 = 1$.

Only knowing if it goes to zero in this limit would be also very interesting; I have physical reason to think it does. This problem appeared in a quantum mechanics problem I'm working on.

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 I couldn't find an asymptotic form for the rhs, and the exact result gets very complicated very quickly with increasing $m$, so I'm stuck.

##Background##

Working on a quantum mechanics problem, I've stumbled on the problem of maximising the functional $$\int_A \varphi_m \varphi_n$$ in the limit of large $m$ and $n$, given that $n \gg m$. $\varphi_m$ 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 rhs eludes me, so

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