A Necessary and Sufficient Discreteness Condition for the Spectrum of a two Term Differential Operator of Higher Order

Abstract

Self-adjoint differential operators of order $2n$ are considered that are associated with the expression $\mathcal{A}\equiv w^{-1}[(—1{)}^n(p{y}^{(n)}{)}^{(n)}+qy]$ and the weighted Hilbert space $L_{2,w}(0,\infty )$. By use of discreteness conditions for the spectrum of such operators given by Kwong and Zettl a necessary and sufficient condition for the discreteness of the spectrum is established.