I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$: $$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$ where $H_m(x)$ is the $m-$th Hermite polynomial. For instance, I would like to understand if $$\frac{J_m}{\sqrt{2^m m!}}=O(m^p)$$ with $p\in\mathbb{R}^−$. Thank you in advance!