We study the asymptotic behaviour of N(α)—the number of negative eigenvalues of the operator (-τ)^l -αV in L₂(R^d) for an even d and 2l≥d. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases where N(α) is of the Weyl-type, N(α) is of the Weyl-order but not the Weyl-type coefficient and finally where N(α)=O(α^q) with q>d/2l.