Asymptotic form of $L^1$-norm of Hermite functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:58:31Z http://mathoverflow.net/feeds/question/65142 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65142/asymptotic-form-of-l1-norm-of-hermite-functions Asymptotic form of $L^1$-norm of Hermite functions Mateus Araújo 2011-05-16T14:26:52Z 2011-06-06T19:42:20Z <h2>Background</h2> <p>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 $\|\varphi_m\|_2 = 1$.</p> <hr> <p>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. </p> <p>Only knowing if it goes to zero in this limit would be also very interesting; I have physical reason to think it does.</p> <p>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</p> <blockquote> <p>Is there an asymptotic form for $\|\varphi_m\|_1$?</p> </blockquote>