The Maslov and Morse indices for Schrodinger operators on [0,1]

TL;DR

This paper establishes a relationship between the Maslov and Morse indices for Schrödinger operators on [0,1], providing an efficient method to compute the Morse index using the Maslov index and matrix eigenvalue problems.

Contribution

It introduces a novel connection between Maslov and Morse indices for Schrödinger operators with symmetric potentials and separated boundary conditions.

Findings

Morse index can be derived from the Maslov index and eigenvalue problems.

Provides an efficient computational approach for Morse index.

Applicable to Schrödinger operators on a finite interval.

Abstract

Assuming a symmetric potential and separated self-adjoint boundary conditions, we relate the Maslov and Morse indices for Schr\"odinger operators on $[0, 1]$. We find that the Morse index can be computed in terms of the Maslov index and two associated matrix eigenvalue problems. This provides an efficient way to compute the Morse index for such operators.