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arXiv · 1509.00707

Geometric aspects of self-adjoint Sturm-Liouville problems

Yicao Wang
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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 $λ_n$ as a function on such orbits.

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Yicao Wang. 2015-09-02. Geometric aspects of self-adjoint Sturm-Liouville problems. https://arxiv.org/abs/1509.00707

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