Weighted Poincare inequality

For a probability density function $w$ on a domain, a weighted Poincare inequality has the formThe optimal $\lambda$ is the spectral gap of the associated weighted diffusion operator.

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