Optimization problems on sums of eigenvalues for Sturm–Liouville operators with <i>L</i> <i>p</i> potentials

This paper presents an equivalent characterization of the optimal estimations for the sums of the first m eigenvalues of Sturm–Liouville operators with Lp potentials. Viewing eigenvalues as nonlinear functionals of potentials, we use the complete continuity of eigenvalues to deduce a family of critical systems consisting of nonlinear ordinary differential equations, which are dependent on the exponent p ∈ (1, ∞) of the Lebesgue spaces and m free parameters. The problem of the optimal estimations for the sums of the first m eigenvalues of Sturm–Liouville operators can be transformed into the solvability of critical systems. Based on this relation, for p = 2 we can theoretically obtain the optimal potential from the integrability theory of Hamiltonian systems point of view. Finally, we observe the numerical experiments for the critical systems to propose the future direction.