Killed Brownian motion and inequalities among solutions of the Schrödinger equation

We construct triplets of killed Brownian motions to obtain inequalities relating the solutions of the Schrödinger equation 12Δψ−hψ=0, with non-negative boundary conditions, on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating harmonic functions on the three compact convex sets and an inequality relating to solutions of the hh-equilibrium potential equations.