  Let $M$ be a complete Riemannian manifold. Let $P_{x,y}(M)$ be the space of
continuous paths on $M$ with fixed starting point $x$ and ending point $y$.
Assume that $x$ and $y$ is close enough such that the minimal geodesic $c_{xy}$
between $x$ and $y$ is unique. Let $-L_{\lambda}$ be the Ornstein-Uhlenbeck
operator with the Dirichlet boundary condition on a small neighborhood of the
geodesic $c_{xy}$ in $P_{x,y}(M)$. The underlying measure
$\bar{\nu}^{\lambda}_{x,y}$ of the $L^2$-space is the normalized probability
measure of the restriction of the pinned Brownian motion measure on the
neighborhood of $c_{xy}$ and $\lambda^{-1}$ is the variance parameter of the
Brownian motion. We show that the generalized second lowest eigenvalue of
$-L_{\lambda}$ divided by $\lambda$ converges to the lowest eigenvalue of the
Hessian of the energy function of the $H^1$-paths at $c_{xy}$ under the small
variance limit (semi-classical limit) $\lambda\to\infty$.
