In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than −ε (ε > 0) of a self-adjoint operator L defined by the following differential expression ℓ(y) = −(p(x)y ′ (x)) ′ − Q(x)y(x) with the boundary condition y(0) = 0 in the space L 2 (0, ∞; H).