# Space versus energy oscillations of Pr\"ufer phases for matrix Sturm-Liouville and Jacobi operators

## TL;DR

This paper establishes a relationship between space oscillations of Pr"ufer phases and eigenvalue counts for matrix Sturm-Liouville and Jacobi operators, supported by theoretical proofs and numerical examples.

## Contribution

It introduces a new oscillation counting method for matrix operators using Pr"ufer phases and proves its equivalence to eigenvalue counting.

## Key findings

- Number of space oscillations equals the number of eigenvalues below a given energy.
- Pr"ufer phases cannot cross -1 in the negative direction, indicating a positivity property.
- Numerical calculations confirm the theoretical results.

## Abstract

This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Pr\"ufer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Pr\"ufer phases, namely they cannot cross $-1$ in the negative direction, and is also shown to be closely linked to the positivity of the matrix Pr\"ufer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example.