## Abstract

A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G = SU(n). On the other hand, there exists a pinned Brownian motion measure ν_{λ} with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □_{λ} on L^{2} (ν _{λ})-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □_{λ} as λ → ∞ by a formal semiclassical analysis.

Original language | English |
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Pages (from-to) | 103-114 |

Number of pages | 12 |

Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics |

Volume | 6 |

Issue number | SUPPL. 1 |

Publication status | Published - 2003 Sep 1 |

## Keywords

- Morse function
- Pinned path group
- Semiclassical analysis
- Witten Laplacian

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Applied Mathematics