Extremal properties of the first eigenvalue of Schr\"odinger-type operators

Given a separable, locally compact Hausdorff space $X$ and a positive Radon measure $m(dx)$ on it, we study the problem of finding the potential $V(x) \ge 0$ that maximizes the first eigenvalue of the Schr\"odinger-type operator $L+V(x)$; $L$ is the generator of a local Dirichlet form $(a, D[a])$ on $L^2(X, m(dx))$.