Green function for Schrödinger operator and conditioned Feynman-Kac gauge

Consider the Schrödinger operator H = (Δ/2) + q in a bounded C1,1 domain D with q ∈ Klocd. (D, q) satisfies the condition: sup[spec((Δ/2) + q)|D] < 0. Let V and G denote the Green functions in D for H and Δ/2, respectively. We prove thatV(x,y)G(x,y)=Eyxexp⁡∫0Tq(xt)dt,x,y∈D,where (xt) is the conditioned Brownian motion starting at x and ending at y with the life time T, and V/G can be extended to a continuous and positive function on &-D × &-D. Specifically, we give the comparison theorem: There exists a constant C = C(D, q) > 0 such that1CG(x,y)⩽V(x,y)⩽CG(x,y),x,y∈D.Another result is that (∂V/∂nz)(x, z), x ∈ D, z ∈ ∂D exists and represents the Poisson kernel for the Dirichlet problem corresponding to Schrödinger operator H