Index Theorems 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 interpretationand generalization of the index theory using the concept of graphical Krein signature. Furthermore, we prove that derivatives of an eigenvector u = u(λ) of an operator pencil L(λ) satisfying L(λ)u(λ) = µ(λ)u(λ) evaluated at a characteristic value of L(λ) do not only generate an arbitrary chain of root vectors of L(λ) but the chain that carries an extra information.

Graphical Krein Signature Theory and Evans--Krein Functions

“…Index theorems similar to those presented here for selfadjoint and alternating (having alternating selfadjoint and skewadjoint coefficients) polynomial matrix pencils of arbitrary degree were recently proved in [42] using advanced results from the algebra of matrix pencils along with perturbation theory for polynomial pencils. Similar results can be found also in [51]. Finally, a very recent and significant development has been the extension of index theorems to cover the case of J = ∂x that includes important examples of traveling waves for Korteweg-deVries-type and Benjamin-Bona-Mahony-type problems.…”

“…In this case ζ can be written in the form N−(D) where D is a Gram-type Hermitian matrix with elements D jk := (Lu [j] , u [k] ) and the vectors u [j] span gKer(JL) Ker(L). Also see [51] for an extensive survey of literature on related index theorems appearing in various fields of mathematics. The relation between the dimension 2nu of the unstable invariant subspace of JL and the number of negative eigenvalues of L was first studied for N−(L) = 1, in which case under the assumption that dim(gKer(JL)) = 2 dim(Ker(L)), the count (85) indicates that JL has at most one pair of non-imaginary (unstable) points of spectrum, and by the full Hamiltonian symmetry these points are necessarily real.…”

“…For the reader familiar with [14,34], the apparent difference between (79) and the formulation of the index theorems found therein is caused by a different choice of the definition of the Krein signature; see the footnote referenced between equations (7) and (8). Finally, note that in [51] the authors give an algebraic formula for the quantity ζ appearing in the statement of Theorem 7 in the case that ζ = Z + (L), and in this formula a key role is played by the canonical set of chains described in Theorem 4.…”