= Ground-state representation of a Sturm-Liouville energy
{c}
{title2=$\int(py'^2+qy^2-\lambda wy^2)=\int pu^2[(y/u)']^2$}
Suppose $u>0$ on an interval solves the <Sturm-Liouville equation> $-(pu')'+qu=\lambda wu$. Writing $y=uv$ and expanding derivatives gives
$$
py'^2+(q-\lambda w)y^2=pu^2v'^2+(puu'v^2)'.
$$
If the endpoint term $[pu'y^2/u]$ vanishes, integration yields the displayed identity. With $p>0$, it proves a lower bound $\lambda$ for the associated <Rayleigh quotient>, attained by $y=u$ when $u$ is admissible. For singular endpoints, establish the identity on compact subintervals and check the boundary limits; this is essential for the <Bessel differential equation> with $p=w=x$, $q=k^2/x$.
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