Asymptotic behavior of the Hermite functions [closed]

I would like to understand the asymptotic behavior of the Hermite function :$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$where $H_k(x)$ is the $k-$th Hermite polynomial. For instance, I would like to understand if for certain fixed points as $x=a\in\mathbb{R}\setminus \mathbb{Q}$ we have$$\psi_k(a)\sim k^{-p},\quad\quad \forall k\in\mathbb{N},$$ with $p>0$. For me, it is actually enough to show that$$\psi_k(a)\gtrsim k^{-p},\quad\quad \forall k\in\mathbb{N}.$$Thank you in advance!