Asymptotics of lowlying Dirichlet eigenvalues of Witten Laplacians on domains in pinned path groups

Abstract

Let $G$ be a compact connected Lie group and $P_{e,a}=C([0,1]\to G~|~γ(0)=e, γ(1)=a)$ be the pinned path space with a pinned Brownian motion measure $ν_{λ,a}$ defined by the heat kernel $p(λ^{-1}t,x,y)$, where $λ$ is a positive parameter. We consider a Witten Laplacian $-L_{λ,\mathcal{D}}$ 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 $a$. $ν_{λ,a}$ has the formal path integral expression $ν_{λ,a}(dγ)=Z_λ^{-1}\exp \left(-λE(γ)\right)dγ$, where $E(γ)=\frac{1}{2}\int_0^1|\dotγ(t)|^2dt$ 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 $-λ^{-1}L_{λ,\mathcal{D}}$ can be approximated by the spectral sets of Ornstein-Uhlenbeck type operators which approximate $-λ^{-1}L_{λ,\mathcal{D}}$ in each small neighborhood of critical points $\{l_i\}$ when $λ\to\infty$. 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 $-λ^{-1}L_{λ\mathcal{D}}$ near the set of the essential spectrum. In this paper, we study the asymptotic behavior of the lowlying discrete spectrum of $-λ^{-1}L_{λ,\mathcal{D}}$ in the complement of the neighborhood of the set of essential spectrum of the approximate Ornstein-Uhlenbeck type operators at $\{l_i\}$.