Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities

We obtain an asymptotic evaluation of the integral $$\int_{\sqrt{2n+1}}^\infty e^{-x^2} H_n^2(x)\,dx$$ for $n\rightarrow\infty$, where $H_n(x)$ is the Hermite polynomial. This integral is used to determine the probability for the quantum harmonic oscillator in the $n$th energy eigenstate to tunnel into the classically forbidden region. Numerical results are given to illustrate the accuracy of the expansion.