Eigenfunctions, deficiency indices and spectra of odd-order differential operators

Abstract

In this paper we determine the asymptotics of the eigenfunctions of a rather general class of odd-order operators on the half line. The coefficients are assumed to satisfy conditions which combine smoothness and decay. The asymptotics of these eigenfunctions allow to compute the deficiency indices of the associated operators. The results are also extended to operators on R, and whenever possible the spectra of the self-adjoint extensions are determined. The form of the eigenfunctions excludes singular continuous spectrum in all cases.