The calculus of Duistermaat's triple index

TL;DR

This paper develops a systematic calculus for the Duistermaat index, a symplectic invariant for triples of Lagrangian subspaces, with applications to eigenvalue problems and relations to other symplectic indices.

Contribution

It provides an axiomatic framework for the Duistermaat index, simplifying proofs of its properties and linking it to other symplectic invariants like the Maslov index.

Findings

Elementary proofs of the index's fundamental properties

Relation of the index to the H"ormander--Kashiwara--Wall and Maslov indices

A formula for the Morse index of Hermitian matrix differences

Abstract

In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention, due to its central role in eigenvalue interlacing problems on quantum graphs (and more abstractly for self-adjoint extensions of symmetric operators). Here we give an axiomatic characterization of the index that leads to elementary proofs of its fundamental properties. We also relate the index to other quantities often appearing in symplectic geometry, such as the H\"ormander--Kashiwara--Wall index and the Maslov index. Among other things, this leads to a curious formula for the Morse index of a difference of Hermitian matrices.