# Morse index theorem

## Statement

### Setup

Let $\R^n$ be Euclidean space, and let $G$ be the linear space of piecewise $C^\infty$-maps from $[0,T]$ to $\R^n$. 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 $\R^n$.

Let $U$ be the unique $C^\infty$-map from $[0,T]\to Hom(\R^n,\R^n)$ 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 $T$.