Morse index theorem

Statement

Setup

Let ${\mathbb{R}}^n$ be Euclidean space, and let $G$ be the linear space of piecewise $C^{\mathrm{\infty }}$-maps from $[0,T]$ to ${\mathbb{R}}^n$. Let $H$ denote the subspace of $G$ comprising maps which are zero at the endpoints (viz maps $u$ satisfying $u(0)=u(T)=0$).

Let $P$ be a $C^{\mathrm{\infty }}$-map from $[0,T]$ into the collection of self-adjoint linear transformations on ${\mathbb{R}}^n$.

Let $U$ be the unique $C^{\mathrm{\infty }}$-map from $[0,T]\to Hom({\mathbb{R}}^n,{\mathbb{R}}^n)$ such that $U^{″}+PU=0$.

Define:

• The multiplicity of $t\in [0,T]$ is the nullity of $U(t)$
• $t\in [0,T]$ is said to be a focal point if its multiplicity is positive

Define the index form corresponding to $P$ as the following bilinear form on $H$:

$I(u,v)=\underset{0}{\overset{T}{\int }}[(u^{\prime },v^{\prime })-(Pu,v)]dt$

Statement part

• The index $i(I)$ (viz the index of the index form) is the sum of multiplicities of all focal points in the open interval $(0,T)$
• The nullity $n(I)$ (viz the nullity of the index form) is the multiplicity of $T$.