Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold

We consider the limiting case λ→0 of the Cauchy problem, ∂gλ(x,t)/∂t = (1/2) λΔxgλ(x,t)+(V(x)/λ) gλ( x,t), with gλ (x,0) = exp{−S0(x)/λ}T0(x), V, S0 being real-valued functions on N, T0 a complex-valued function on N; V, S0, T0 being independent of λ, Δx being the Laplace–Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem D2Z(s)/∂s2 = −∇ZV[Z(s)], s∈[0,t], with Z(t) = x and Z(0) = ∇S0(Z(0)).One of our results is equivalent to the fact that for short times Schrödinger quantum mechanics on the Riemannian manifold N tends to classical Newtonian mechanics on N as h/ tends to zero.