Fredholm-Lagrangian-Grassmannian and the Maslov index

Abstract

We explain the topology of the space, so called, Fredholm-Lagrangian-Grassmannain and the quantity ``Maslov index'' for paths in this space based on the standard theory of Functional Analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredholm operators, which was first recognized by J. Phillips and used to define the ``Spectral flow''. We tried to make the arguments to be all elementary and we summarize basic facts for this article from Functional Analysis in the Appendix.

64 pages, no figure

References in corpus (2)

• A general splitting formula for the spectral flow
• On the Quillen determinant

Cited by in corpus (15)

• The Maslov Index and the Spectra of Second Order Elliptic Operators
• Instability of pulses in gradient reaction-diffusion systems: A symplectic approach
• Lagrangian Grassmannian in Infinite Dimension
• The Morse and Maslov Indices for Schrödinger Operators
• The Maslov and Morse indices for Schrödinger operators on $\mathbb{R}$
• Smooth Fock bundles, and spinor bundles on loop space
• Partial Signatures and the Yoshida-Nicolaescu Theorem
• An index theorem for Schrödinger operators on metric graphs
• A Morse index theorem for elliptic operators on bounded domains
• The Maslov index for Lagrangian pairs on ${\mathbb{R}}^{2n}$
• First-order asymptotic perturbation theory for extensions of symmetric operators
• Signature and spectral flow of $J$-unitary $S^1$-Fredholm operators
• The positronium and the dipositronium in a Hartree-Fock approximation of quantum electrodynamics
• The restricted Lagrangian Grassmannian in infinite dimension
• On the Homotopy Type of the Fredholm Lagrangian Grassmannian