Uniqueness of a Negative Mode About a Bounce Solution

TL;DR

This paper investigates the uniqueness of a negative eigenvalue in the spectrum of fluctuations around a bounce solution in multiple dimensions, using Morse theory and conjugate points to establish the result.

Contribution

It extends the nodal theorem approach to multidimensional bounce solutions, demonstrating the existence of exactly one conjugate point at a0=0 with multiplicity one.

Findings

Bounce solution has exactly one conjugate point at a0=0

The approach generalizes the nodal theorem to higher dimensions

Establishes uniqueness of the negative eigenvalue in the spectrum

Abstract

We consider the uniqueness problem of a negative eigenvalue in the spectrum of small fluctuations about a bounce solution in a multidimensional case. Our approach is based on the concept of conjugate points from Morse theory and is a natural generalization of the nodal theorem approach usually used in one dimensional case. We show that bounce solution has exactly one conjugate point at $\tau=0$ with multiplicity one.