Observe Sturm-Liouville problem on entire line $$-(p(x)y'(x))' + l(x)y(x)= \lambda r(x)y(x), \hspace{3mm} -\infty<x<\infty \tag{1} \label{1}$$ where $p(x)$ and $r(x)$ are positive on $\mathbb{R}$. Assume $p(x),r(x),l(x) \in C^{2}(\mathbb{R})$ (maybe it is enough $p(x)$ has a contionous first derivate and $p(x)r(x)$ has continous second derivate).
$\mathbf{Question:}$ How to reduce $\eqref{1}$ to the form
$$-u''(x) + q(x)u(x)= \mu u(x) $$
By Levitan, Sargasjan - Introduction to spectral theory,
substitution on finite interval $[a,b]$ is given
$$z=\frac{1}{c} \int_{a}^{x} \bigg(\frac{r(x)}{p(x)}\bigg)^{1/2} dx, \hspace{3mm}u=(r(x)p(x))^{1/4}y(x),\hspace{3mm} \mu=c \lambda,$$
where c is given by $$c=\frac{1}{\pi}\int_{a}^{b} \bigg(\frac{r(x)}{p(x)}\bigg)^{1/2}dx$$ and $$q(z)=\frac{\theta''(z)}{\theta(z)}-c^2 \frac{l(x)}{r(x)}, \hspace{3mm} \theta(z)=(r(x)p(x))^{1/4}$$
As the result of this change the interval $[a,b]$ is transformed to the interval $[0,\pi]$.
Is it same for whole line $\mathbb{R}$?
