The dual eigenvalue problems for the Sturm–Liouville system

In this paper, we find the minimizer of the eigenvalue gap for the Schrödinger equation and vibrating string equation. In the first part, we show the first two Neumann eigenvalue gap of the Schrödinger equation with single-well potentials is not less than 1 and the equality holds if and only if the potential is constant. In the second part, since the first Neumann eigenvalue of the vibrating string equation is 0, we turn to show that the minimizing density function of the second Neumann eigenvalue is of the form hχ(a,π−a)+Hχ[0,π]\(a,π−a) for some a.