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

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) $\lambda\to\infty$.