We give estimates for positive solutions for the Schrödinger equation $(Δ_μ+W)u=0$ on a wide class of parabolic weighted manifolds $(M, dμ)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan to give matching upper and lower bounds for the heat kernel of the corresponding Schrödinger operator $Δ_μ+W$. In particular, this appears to complement known results for Schrödinger operators on $\mathbf{R}^2$.

30 pages