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} ( G ) = C ([0, 1] → 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

acting on functions with the Dirichlet boundary condition on a certain domain

D ⊂ P e , a ( G ) $\mathcal{D}\subset {P}_{e,a}\left(G\right)$

which includes finitely many geodesics { l ₁ , …, l _N } between e and a . ν _{λ , a} has the formal path integral expression

ν λ , a ( d γ ) = Z λ − 1 ⁡ exp − λ E ( γ ) d γ ${\nu }_{\lambda ,a}\left(\mathrm{d}\gamma \right)={Z}_{\lambda }^{-1}\mathrm{exp}\left(-\lambda E\left(\gamma \right)\right)\mathrm{d}\gamma $

, where

E ( γ ) = 1 2 ∫ 0 1 | γ ̇ ( t ) | 2 d t $E\left(\gamma \right)=\frac{1}{2}{\int }_{0}^{1}\vert \dot {\gamma }\left(t\right){\vert }^{2}\mathrm{d}t$

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 , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$

can be approximated by the spectral sets of Ornstein-Uhlenbeck type operators which approximate

− λ − 1 L λ , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$

in each small neighborhood of critical points { l _i } when λ → ∞ . 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 , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$

near the set of the essential spectrum. In this paper, we study the asymptotic behavior of the lowlying discrete spectrum of

− λ − 1 L λ , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$

in the complement of the neighborhood of the set of essential spectrum of the approximate Ornstein-Uhlenbeck type operators at { l _i } as  λ  → ∞.