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

TL;DR

This paper investigates the maximization of the sum of the first two eigenvalues of Sturm-Liouville operators with potentials in L^1, proving existence, uniqueness, and characterizing the maximizer through differential equations.

Contribution

It establishes the existence and uniqueness of the maximizing potential and characterizes it via solutions to the pendulum differential equation.

Findings

Unique potential maximizes the sum of the first two eigenvalues.

Maximizer is non-negative, piecewise smooth, and symmetric.

The maximizer's nonzero part is described by the pendulum equation.

Abstract

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$^*$ convergence, we show that the nonzero part of the maximizer can be determined by the solution to the pendulum equation $\theta'' + \ell \sin\theta = 0 $.