Ground-state transform for a weighted Dirichlet energy

Let $\Phi \in C^2(I)$ on an open interval $I$, take a real smooth function $h$ of compact support in $I$, and put $w=e^{-\Phi }$ and $g=h{e}^{-\Phi /2}$. Expanding $h^{\prime }{e}^{-\Phi /2}=g^{\prime }+\frac{1}{2}{\Phi }^{\prime }g$ and using integration by parts givesThis transforms the weighted gradient energy into the quadratic form of a Schrödinger operator with potential $|{\Phi }^{\prime }{|}^2/4-{\Phi }^{\prime \prime }/2$.

On a finite interval $[a,b]$, without boundary vanishing the right-hand side additionally containsThus the boundary-free identity requires compact support, suitable zero boundary conditions, or sufficient endpoint decay. The corresponding infinite-interval statement also requires convergence of the integrals and disappearance of the boundary term.