We give estimates for positive solutions for the Schro & uml;dinger equation (z mu + W)u = 0 on a wide class of parabolic weighted manifolds (M, d & micro;) when W decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan [Heat kernels on weighted manifolds and applications, The ubiquitous heat kernel, Amer. Math. Soc., Providence, RI, 2006] to give matching upper and lower bounds for the heat kernel of the corresponding Schro & uml;dinger operator z mu + W. In particular, this appears to complement known results for Schro & uml;dinger operators on R2.
