Jacobi identity for the second variation ($P{\phi }^{\prime 2}+Q{\phi }^2=P({\phi }^{\prime }-w^{\prime }\phi /w{)}^2+(P{w}^{\prime }{\phi }^2/w{)}^{\prime }$)

If $w$ is a nonvanishing solution of $-(P{w}^{\prime }{)}^{\prime }+Qw=0$, expanding the right-hand side gives the displayed identity. Integrating expresses a scalar second variation as a square plus a boundary term. A simple zero of $w$ at a fixed endpoint causes no difficulty for Lipschitz endpoint-vanishing variations, since $\phi =O(x-a)$ makes the boundary term tend to zero. This identity is the scalar form of a ground-state transform for a weighted Dirichlet energy.