Index Theorems for Polynomial Pencils

TL;DR

This paper reviews index theorems for polynomial operator pencils, introduces a graphical interpretation via Krein signatures, and proves new properties of eigenvector derivatives at characteristic values.

Contribution

It provides a unified graphical framework for index theory and reveals new insights into eigenvector derivatives at characteristic values.

Findings

Graphical Krein signature offers a simple interpretation of index theorems.

Eigenvector derivatives at characteristic values encode additional structural information.

The paper generalizes existing index theory for polynomial pencils.

Abstract

We survey index theorems counting eigenvalues of linearized Hamiltonian systems and characteristic values of polynomial operator pencils. We present a simple common graphical interpretation and generalization of the index theory using the concept of graphical Krein signature. Furthermore, we prove that derivatives of an eigenvector u= u(\lambda) of an operator pencil L(\lambda) satisfying L(\lambda) u(\lambda)= \mu(\lambda) u(\lambda) evaluated at a characteristic value of L(\lambda) do not only generate an arbitrary chain of root vectors of L(\lambda) but the chain that carries an extra information.