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arXiv · 2509.09250

Non-integrability of the Critical Systems for Optimal Sums of Eigenvalues of Sturm-Liouville Operators

Yuzhou Tian·Meirong Zhang
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Abstract

The optimal lower or upper bounds for sums of the first $m$ eigenvalues of Sturm-Liouville operators can be obtained by solving the corresponding critical systems, which are Hamiltonian systems of $m$ degrees of freedom with $m$ parameters. With the help of the differential Galois theory, we prove that these critical systems are not meromorphic integrable except for two known completely integrable cases. The non-integrability of the critical systems reveal certain complexities for the original eigenvalues problems.

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Yuzhou Tian, Meirong Zhang. 2025-09-11. Non-integrability of the Critical Systems for Optimal Sums of Eigenvalues of Sturm-Liouville Operators. https://arxiv.org/abs/2509.09250

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