Optimal maximal gaps of Dirichlet eigenvalues of Sturm–Liouville operators

Abstract

In this paper, we consider the gaps λ2 n( q) − λ1( q) for the Dirichlet eigenvalues { λ m( q)} of Sturm–Liouville operators with potentials q on the unit interval. By merely assuming that potentials q have the L1 norm  r, we will explicitly give the solutions to the maximization problems of λ2 n( q) − λ1( 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)].