# On the $j$-th Eigenvalue of Sturm-Liouville Problem and the Maslov Index

## TL;DR

This paper establishes a precise relationship between the jump phenomena of the $j$-th eigenvalue in Sturm-Liouville problems and the Maslov index, providing new insights into eigenvalue behavior and boundary condition limits.

## Contribution

It demonstrates that the jump number of eigenvalue branches equals the Maslov index and determines the eigenvalue range for boundary condition layers.

## Key findings

- Jump number equals Maslov index for boundary conditions
- Sharp eigenvalue range on boundary condition layers
- Monodromy matrix approaches Dirichlet condition as spectral parameter decreases

## Abstract

In the previous papers \cite{HLWZ,KWZ}, the jump phenomena of the $j$-th eigenvalue were completely characterized for Sturm-Liouville problems. In this paper, we show that the jump number of these eigenvalue branches is exactly the Maslov index for the path of corresponding boundary conditions. Then we determine the sharp range of the $j$-th eigenvalue on each layer of the space of boundary conditions. Finally, we prove that the graph of monodromy matrix tends to the Dirichlet boundary condition as the spectral parameter goes to $-\infty$.