The Morse and Maslov indices for matrix Hill's equations

TL;DR

This paper explores the relationship between Morse and Maslov indices in matrix Hill's equations with periodic potentials, adapting multidimensional strategies to a one-dimensional setting.

Contribution

It introduces a novel approach to relate Morse and Maslov indices specifically for matrix Hill's equations with periodic potentials.

Findings

Established a connection between Morse and Maslov indices in this setting.

Adapted multidimensional index relations to one-dimensional periodic problems.

Provides a framework for analyzing stability in matrix Hill's equations.

Abstract

For Hill's equations with matrix valued periodic potential, we discuss relations between the Morse index, counting the number of unstable eigenvalues, and the Maslov index, counting the number of signed intersections of a path in the space of Lagrangian planes with a fixed plane. We adapt to the one dimensional periodic setting the strategy of a recent paper by J. Deng and C. Jones relating the Morse and Maslov indices for multidimensional elliptic eigenvalue problems.