Geometric aspects of self-adjoint Sturm-Liouville problems

TL;DR

This paper explores the geometric structure of self-adjoint boundary conditions in Sturm-Liouville problems using the group U(2), classifies these conditions into orbits, and studies how eigenvalues vary within these classes.

Contribution

It introduces a geometric framework using U(2) to classify boundary conditions and explicitly parameterizes principal orbits, analyzing eigenvalue behavior on these orbits.

Findings

Explicit parameterizations of principal orbits as 2-spheres.

Eigenvalue functions exhibit specific behaviors on these orbits.

Refined classification of boundary conditions via group actions.

Abstract

In the paper, we use $\mathrm{U}(2)$, the group of $2\times 2$ unitary matices, to parameterize the space of all self-adjoint boundary conditions for a fixed Sturm-Liouville equation on the interval $[0,1]$. The adjoint action of $\mathrm{U}(2)$ on itself naturally leads to a refined classification of self-adjoint boundary conditions--each adjoint orbit is a subclass of these boundary conditions. We give explicit parameterizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere $S^2$, and investigate the behavior of the $n$-th eigenvalue $\lambda_n$ as a function on such orbits.