Ground-state representation of a Sturm-Liouville energy ($\int (p{y}^{\prime 2}+q{y}^2-\lambda w{y}^2)=\int p{u}^2[(y/u{)}^{\prime }{]}^2$)

Suppose $u>0$ on an interval solves the Sturm-Liouville equation $-(p{u}^{\prime }{)}^{\prime }+qu=\lambda wu$. Writing $y=uv$ and expanding derivatives givesIf the endpoint term $[p{u}^{\prime }{y}^2/u]$ 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 $p=w=x$, $q=k^2/x$.