The Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$

TL;DR

This paper establishes a precise relationship between the Maslov and Morse indices for Schrödinger operators on the real line with symmetric, asymptotically constant potentials, showing they are negatives of each other under certain conventions.

Contribution

It provides a rigorous link between Maslov and Morse indices for Schrödinger operators with specific asymptotic conditions, clarifying their equivalence up to a sign.

Findings

Morse index equals the negative of the Maslov index under chosen conventions.

The relationship holds for symmetric potentials with sufficient asymptotic decay.

The results deepen understanding of spectral properties of Schrödinger operators.

Abstract

Assuming a symmetric potential that approaches constant endstates with a sufficient asymptotic rate, we relate the Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index.