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

We consider the limiting case ??0 of the Cauchy problem,?g_?(x,t)/?t = 1/2?? _xg_?(x,t) + V(x)/?g_?(x,t), with g_?(x,O) = exp {-S_O(x)/? } T _O(x); V, S_O being real-valued functions on N;T_O a complex-valued function on N; V, S_O, T_O being independent of ?;?_x 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)/?s ² = -?_ZV[Z(s)], se[O,t], with Z(t) = x and Z (O) = ?S_O(Z (O)). 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 ? tends to zero. © 1981 American Institute of Physics.