We investigate the heat kernels of Schrödinger operators in both Euclidean space and on weighted Riemannian manifolds. We obtain upper and lower bounds for these heat kernels when the potential of the Schrödinger operator decays faster than quadratically at infinity, in a suitable sense. Additionally, we investigate the 3G Principle in fractal-type spaces, and show that it is equivalent to the Boundary Harnack Principle for a large family of such spaces.