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Let $M$ be a complete Riemannian manifold. Let $P_{x,y}(M)$ be the space ofcontinuous 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-Uhlenbeckoperator with the Dirichlet boundary condition on a small neighborhood of thegeodesic $c_{xy}$ in $P_{x,y}(M)$. The underlying measure$\bar{\nu}^{\lambda}_{x,y}$ of the $L^2$-space is the normalized probabilitymeasure of the restriction of the pinned Brownian motion measure on theneighborhood of $c_{xy}$ and $\lambda^{-1}$ is the variance parameter of theBrownian motion. We show that the generalized second lowest eigenvalue of$-L_{\lambda}$ divided by $\lambda$ converges to the lowest eigenvalue of theHessian of the energy function of the $H^1$-paths at $c_{xy}$ under the smallvariance limit (semi-classical limit) $\lambda\to\infty$.

Semi-classical limit of the generalized second lowest eigenvalue of Dirichlet Laplacians on small domains in path spaces