Asymptotics of the number of eigenvalues of one-term second-order operator equations

We study one-term operator L acting in the space $H_{1}=L_{2}([0,\infty );H)$ generated by the operator-differential expression $\mathcal{L}=-\frac{d}{dx} ( P(x)\frac{d}{dx})$ and the boundary condition $y(0)=0$ . We evaluate the asymptotic number of eigenvalues of the operator L under certain conditions.