Morse index theorem

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Template: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.