Asymptotics of lowlying Dirichlet eigenvalues of Witten Laplacians on domains in pinned path groups
Let $G$ be a compact connected Lie group and $P_{e,a}=C([0,1]\to G~|~\gamma(0)=e, \gamma(1)=a)$ be the pinned path space with a pinned Brownian motion measure $\nu_{\lambda,a}$ defined by the heat kernel $p(\lambda^{-1}t,x,y)$, where $\lambda$ is a positive parameter. We consider a Witten Laplacian $-L_{\lambda,\mathcal{D}}$ acting on functions with the Dirichlet boundary condition on a certain domain $\mathcal{D}\subset P_{e,a}(G)$ which includes finitely many geodesics $\{l_1,\ldots,l_N\}$ between $e$ and $a$. $\nu_{\lambda,a}$ has the formal path integral expression $\nu_{\lambda,a}(d\gamma)=Z_{\lambda}^{-1}\exp \left(-\lambda E(\gamma)\right)d\gamma$, where $E(\gamma)=\frac{1}{2}\int_0^1|\dot{\gamma}(t)|^2dt$ and $E$ is a Morse function when $a$ is not a point of the cut-locus of $e$. Hence, by the analogy of finite dimensional cases, one may expect that the lowlying spectrum of $-\lambda^{-1}L_{\lambda,\mathcal{D}}$ can be approximated by the spectral sets of Ornstein-Uhlenbeck type operators which approximate $-\lambda^{-1}L_{\lambda,\mathcal{D}}$ in each small neighborhood of critical points $\{l_i\}$ when $\lambda\to\infty$. However, in contrast to the finite dimensional case, the spectral sets of the approximate Ornstein-Uhlenbeck type operators contain essential spectrum. It may be difficult to analyze the behavior of the spectrum of $-\lambda^{-1}L_{\lambda\mathcal{D}}$ near the set of the essential spectrum. In this paper, we study the asymptotic behavior of the lowlying discrete spectrum of $-\lambda^{-1}L_{\lambda,\mathcal{D}}$ in the complement of the neighborhood of the set of essential spectrum of the approximate Ornstein-Uhlenbeck type operators at $\{l_i\}$.