Ground-state transform for a weighted Dirichlet energy

For $w=e^{-\Phi }$ and $g=h{e}^{-\Phi /2}$, integration by parts givesThis conjugates the weighted Dirichlet energy to a Schrödinger operator with effective potential $|{\Phi }^{\prime }{|}^2/4-{\Phi }^{\prime \prime }/2$.