Characterization of Maximizers for Sums of the First Two Eigenvalues of Sturm-Liouville Operators

In this paper we study the maximization of the sum of the first two Dirichlet eigenvalues for Sturm-Liouville operators with potentials in the noncompact space $L^1$. We prove that there exists a unique potential function achieving the maximum, which is non-negative, piecewise smooth, and symmetric. Using measure differential equations and weak${}^{\ast }$ convergence, we show that the nonzero part of the maximizer can be determined by the solution to the pendulum equation ${\theta }^{\mathrm{\prime }\mathrm{\prime }}+\mathrm{\ell }\sin \theta =0$.