Extremal eigenvalue gaps for the Schrödinger operator with Dirichlet boundary conditions

We consider the problems of minimizing and maximizing the gap between the two lowest Dirichlet eigenvalues of the Schrödinger operator -Δ + V(x) on a bounded domain in Rⁿ, when the potential V is subjected to a p-norm constraint. We give characterization theorems for extrenming potentials. We prove in particular that a second eigenvalue corresponding to a minimizing potential is always single.