Reading Week Bonus problem 4

[Victor Ivrii]

Using Reading Week Bonus problem 3 find inequalities for two consecutive zeros $x_n$ and $x_{n+1}$ of Airy function satisfying equation
\begin{equation}
y''+xy=0,
\end{equation}
Then derive asymptotic formula for $x_n$ as $n\to +\infty$.

[Victor Ivrii]

Applying Problem 3 we conclude that solution does not oscillate on $(-\infty,0]$ and that on $(0,\infty)$ 
$x_{n+1}-x_n \sim \frac{\pi}{\sqrt{x_n}}$ and $x_n\sim x_{n+1}$ for $n\gg 1$.

Then
\begin{equation*}
x_{n+1}^{\frac{3}{2}}-x_n^{\frac{3}{2}}\sim \frac{3}{2}\pi \implies x_n^{\frac{3}{2}} \sim \frac{3}{2}\pi n \implies x_n\sim  \bigl(\frac{3}{2}\pi n\bigr)^{\frac{2}{3}}.
\end{equation*}

Remark. One can prove that "amplitude" decays as $|x|^{-\frac{1}{4}}$ as $x\to +\infty$. Almost all Airy functions grow fast as $x\to -\infty$ with exception of one (up to a constant factor) which is fast decaying as $x\to -\infty$.

Remark. Airy functions play important role in the study of the high frequency electromagnetic field
* near simple caustics
* near diffraction point