Why the below holds for $n\geq 1$? $$\frac{(-i)^n}{\sqrt{2\pi}} \int_{-\infty}^\infty \sqrt{x^2+p^2} e^{ipx} H_n(p) e^{-p^2/2} dp \approx \sqrt{2n+1} H_n(x) e^{-x^2/2}.$$
The Hermite polynomial is defined as $$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}.$$
I have a numerical evidence that LHS and RHS are very similar for $n\geq 1$. For $n=0,\ldots, 4$, I computed the both sides numerically, and obtained the following:
The blue line is LHS, and the orange line is RHS. They agree quite well, but there is a small difference. Indeed, at $x=0$, we can compute the both sides analytically, and there was some (~10%) discrepancy. Still, why there are approximately equal is a mystery.