Morse index theorem

Statement

Setup

Let $\mathbb {R} ^{n}$ be Euclidean space, and let $G$ be the linear space of piecewise $C^{\infty }$-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 $C^{\infty }$-map from $[0,T]$ into the collection of self-adjoint linear transformations on $\mathbb {R} ^{n}$.

Let $U$ be the unique $C^{\infty }$-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 $H$:

$I(u,v)=\int _{0}^{T}[(u',v')-(Pu,v)]dt$

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 $T$.