On the Meromorphic Integrability of the Critical Systems for Optimal Sums of Eigenvalues

Abstract: The popularity of estimation to bounds for sums of eigenvalues started from P. Li and S. T. Yau for the study of the P\'{o}lya conjecture. This subject is extended to different types of differential operators. This paper explores for the sums of the first $m$ eigenvalues of Sturm-Liouville operators from two aspects. Firstly, by the complete continuity of eigenvalues, we propose a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent $p\in(1,\infty)$ of the Lebesgue spaces concerned. There have profound relations between the solvability of these systems and the optimal lower or upper bounds for the sums of the first $m$ eigenvalues of Sturm-Liouville operators, which provides a novel idea to study the optimal bounds. Secondly, we investigate the integrability or solvability of the critical systems. With suitable selection of exponents $p$, the critical systems are equivalent to the polynomial Hamiltonian systems of $m$ degrees of freedom. Using the differential Galois theory, we perform a complete classification for meromorphic integrability of these polynomial critical systems. As a by-product of this classification, it gives a positive answer to the conjecture raised by Tian, Wei and Zhang [J. Math. Phys. 64, 092701 (2023)] on the critical systems for optimal eigenvalue gaps. The numerical simulations of the Poincar\'{e} cross sections show that the critical systems for sums of eigenvalues can appear complex dynamical phenomena, such as periodic trajectories, quasi-periodic trajectories and chaos.