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Hessian formulas and estimates for parabolic Schrödinger operators

We study the Hessian of the fundamental solution to the parabolic problem for weighted Schrödinger operators of the form $\frac 12 Δ+\nabla h-V$ proving a second order Feynman-Kac formula and obtaining Hessian estimates. For manifolds with a pole, we use the Jacobian determinant of the exponential map to offset the volume growth of the Riemannian measure and use the semi-classical bridge as a delta measure at $y_0$ to obtain exact Gaussian estimates. These estimates are in terms of bounds on $Ric-2 Hess (h)$, on the curvature operator, and on the cyclic sum of the gradient of the Ricci tensor.