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In this work, we find the asymptotic formulas for the sum of the negativeeigenvalues smaller than $-\varepsilon$ $(\varepsilon >0)$ of a self-adjointoperator $L$ which is defined by the following differential expression$$\ell(y)=-(p(x)y'(x))'-Q(x)y(x)$$ with the boundary condition $y(0) = 0$ inthe space in the space $L_{2}(0,\infty ;H)$.

The Asymptotic Behaviour of the Sum of Negative Eigenvalues of a Self-Adjoint Operator Given in Semi-Axis