WKB form substitution for V

Question

If I have the equation $$y'' +\sin(1/x)y = 0$$ how would I replace $\sin(1/x)y$ in order to get epsilon squared in front of the first term? I have tried defining $X= \delta\sin(1/x)$, but that doesn't help at all. Any help would be much appreciated. Thanks!

Answer 1

At least for large $x$ we can approximate $\sin(1/x)\approx 1/x$ to get

$$y''(x)+\frac{1}{x}y(x)=0$$

Which can be solved in terms of Bessel functions: $$y(x)=c_1 \sqrt{x}~ J_1(2\sqrt{x})+c_2\sqrt{x}~Y_1(2\sqrt{x})$$

They admit asymptotic expansions for real $x$ $$J_\nu(x)=\sqrt{\frac{2}{\pi x}}\left[\cos\left(x-\frac{\nu \pi}{2}-\frac{\pi}{4}\right)+O(|x|^{-1})\right] $$

$$Y_\nu(x)=\sqrt{\frac{2}{\pi x}}\left[\sin\left(x-\frac{\nu \pi}{2}-\frac{\pi}{4}\right)+O(|x|^{-1})\right] $$

Meaning in our case

$$\sqrt{x}~J_1(2\sqrt{x})\asymp \sqrt{\frac{2}{\pi}}x^{1/4}\left[\cos\left(2\sqrt{x}-\frac{3 \pi}{4}\right)\right]$$

If we give the two equations identical initial conditions at $x=1$ we can see the fit is already quite good:

I'm afraid however around the origin the behavior is extremely hard to model due to the very fast oscillation of the sine term. Hopefully this is a good start.