Optimal maximal gaps of Dirichlet eigenvalues of Sturm-Liouville operators

In this paper, we consider the gaps lambda(2n)(q) - lambda(1)(q) for the Dirichlet eigenvalues {lambda(m)(q)} of Sturm-Liouville operators with potentials q on the unit interval. By merely assuming that potentials q have the L-1 norm r, we will explicitly give the solutions to the maximization problems of lambda(2n)(q) - lambda(1)(q), where n is arbitrary. As a consequence, the solutions can lead to the optimal upper bounds for these eigenvalue gaps. The proofs are extensively based on the eigenvalue theory of measure differential equations in Meng and Zhang [J. Differ. Equations 254, 2196-2232 (2013)] and on the known results of the optimization problems for single eigenvalues of ordinary differential equations in Wei, Meng, and Zhang [J. Differ. Equations 247, 364-400 (2009)]. Published under an exclusive license by AIP Publishing.