The Maslov index for Lagrangian pairs on $\mathbb{R}^{2n}$

TL;DR

This paper introduces a spectral flow-based definition of the Maslov index for Lagrangian pairs in n, exploring its properties and applications to relate it to the Morse index in Schrodinger operators.

Contribution

It provides a new spectral flow-based approach to defining the Maslov index for Lagrangian pairs and demonstrates its utility in analyzing Morse indices of Schrodinger operators.

Findings

The spectral flow approach simplifies the analysis of the Maslov index.

Established a clear relationship between the Maslov and Morse indices.

Applied the method to Schrodinger operators on different domains.

Abstract

We discuss a definition of the Maslov index for Lagrangian pairs on $\mathbb{R}^{2n}$ based on spectral flow, and develop many of its salient properties. We provide two applications to illustrate how our approach leads to a straightforward analysis of the relationship between the Maslov index and the Morse index for Sch\"odinger operators on $[0,1]$ and $\mathbb{R}$.