Spectral flow, Brouwer degree and Hill's determinant formula

TL;DR

This paper generalizes a topological invariant related to conjugate points in semi-Riemannian geometry to Morse-Sturm systems, establishes a spectral flow formula, and explores its connection to Hill's determinant, with applications to Hamiltonian system stability.

Contribution

It introduces a new spectral flow formula for Morse-Sturm systems and relates it to Hill's determinant, extending previous invariants for conjugate point counting.

Findings

Derived a spectral flow formula for Morse-Sturm systems

Connected the spectral flow to Hill's determinant formula

Applied the invariant to detect Hamiltonian system instability

Abstract

In 2005 a new topological invariant defined in terms of the Brouwer degree of a determinant map, was introduced by Musso, Pejsachowicz and the first name author for counting the conjugate points along a semi-Riemannian geodesic. This invariant was defined in terms of a suspension of a complexified family of linear second order Dirichlet boundary value problems. In this paper, starting from this result, we generalize this invariant to a general self-adjoint Morse-Sturm system and we prove a new spectral flow formula. Finally we discuss the relation between this spectral flow formula and the Hill's determinant formula and we apply this invariant for detecting instability of periodic orbits of a Hamiltonian system.