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

Abstract

Let $G$ be a compact connected Lie group and $P_{e,a}=C([0,1]\to G|\gamma (0)=e,\gamma (1)=a)$ 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 $\{l_1,\dots ,l_N\}$ 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 $E(\gamma )=\frac{1}{2}{\int }_0^1|\dot{\gamma }(t){|}^{2}dt$ 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 $\{l_i\}$.

arXiv commentThis modified version will appear in Advanced nonlinear studies. 92 Pages