Аннотация:Necessary and sufficient conditions are obtained for the discreteness of thespectrum of the operator Lu(x):= 1 ρ(x) (−1)m(p(x)u(m)(x))(m), x∈I=[0,∞), m≥1. In the case of p(x)=x^ν,ν∈[0,1),they are the same: lim s→∞ s^{2m−1−ν} \int_s^\infty ρ(x)dx=0.