Regular and Pathological Eigenvalue Behavior for the Equation −λu″=Vuon the Semiaxis

We study the behavior of the eigenvalue distribution functionn(λ) for the equation−λu″=Vu on R+, u(0)=0.Typically,n(λ)=O(λ−1/2) and, moreover,λ1/2n(λ)→π−1 ∫ V dxasλ→0. We obtain criteria for each of these relations. Examples are presented where limλ→0 λ1/2n(λ) does exist, but is not equal toπ−1 ∫ V dx.