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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 $\mathbb{R}$?

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  • $\begingroup$ I do not see what would prevent the same transformation to hold on the whole line except when the integral $c$ diverges. However, the same problem for the finite interval presents itself if you allow singularities especially at the endpoints of $[a,b]$, doesn't it? $\endgroup$
    – Hans
    Jan 11, 2018 at 21:18
  • $\begingroup$ Is there any other substitution? $\endgroup$ Jan 11, 2018 at 21:41
  • $\begingroup$ You could ask the same question regarding other substitutions on the $[a,b]$, couldn't you? How does it distinguish $[a,b]$ from the whole real line? $\endgroup$
    – Hans
    Jan 11, 2018 at 22:55

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