Bounds on Green’s functions of second-order differential equations

We estimate the diagonal part of the Green’s function for the equation (−Δ/2+V(x)+∂/∂t)ψ(x, t) = 0, t≳0, x∈ B, where B is a finite region of the Euclidean space Rd with a regular boundary. In the special case V(x) = 0, x∈B, we also obtain bounds for the nondiagonal part of the Green’s function which are uniform in t.