[Submitted on 10 Dec 2025 (v1), last revised 4 Jul 2026 (this version, v2)]

View PDF

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 $\{l_1,\ldots,l_N\}$ 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 $\{l_i\}$.

Submission history

From: Shigeki Aida [view email]
[v1] Wed, 10 Dec 2025 08:32:36 UTC (99 KB)
[v2] Sat, 4 Jul 2026 13:47:01 UTC (80 KB)

Current browse context:

math.PR

Bookmark

Bibliographic and Citation Tools

Code, Data and Media Associated with this Article

Demos

Recommenders and Search Tools

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.