Every Sturm-Liouville problem is a Schrödinger equation

Question

Imagine the Sturm-Liouville problem $$ (pu')'+qu+\lambda ru=pu''+p'u'+qu+\lambda ru=0 $$ Let $h=\frac{1}{(pq)^{\frac{1}{4}}}$, $u=hv$. Deriving in the above equation and multiplying by h, we arrive at $$ h(p(hv''+2h'v'+h''v)+p'(hv'+h'v)+qhv+\lambda rhv)=0 $$ $$ (h^2pv')'+h(ph''+p'h'+qh)v+\lambda r h^2 v=0 $$ Let $\hat{p}=h^2p$, $\hat{q}=h(ph''+p'h'+qh)$, $\hat{r}=rh^2$. Thus $\hat{r}\hat{p}=1$. Now let $y=\int\frac{1}{\hat{p}(x)}dx = \int\hat{r}(x)dx$. Now, let $w(y)=v(x)$. Thus, $w$ verifies $$ w''+\frac{\hat{q}}{\hat{p}}w+\lambda v=0 $$ Let $\tilde{q}=\frac{\hat{q}}{\hat{p}}(x(y))$. Thus, equation has the form $$ w''+\tilde{q}w+\lambda v=0 $$ which is Schrödinger equation. Anywhere where I can find more about this?