Killed Random Processes and Heat Kernels

Let $V(x)\geq0$ be a given function tending to a constant at infinity. It is well known that the density of the Brownian motion $B_t$ killed at the infinitesimal rate $V$ is a Green's function for the heat operator with such a potential. With an appropriate generalization, its Laplace transform also gives the density of $\int_0^tV(B_s)ds$. We construct such a Green's function via spectral analysis of the classical one-dimensional stationary Schrodinger operator.