Asymptotics of the eigenvalues of the Dirichlet and Neumann problems for the abstract Sturm-Liouville operator

Let $A>0$ be an unbounded self-adjoint operator in a Hilbert space $H$. In the Hilbert space $H_1=L_2(0,\pi;H)$ we study the spectrum of the differential equations
\begin{gather*} -y''(x)+Ay=\lambda y,\quad y(0)=y(\pi)=0, \\ -y''(x)+Ay=\lambda y,\quad y'(0)=y'(\pi)=0. \end{gather*}
We find the principal terms of the asymptotics of the functions $N(\lambda)$ for these problems and we ascertain the conditions under which they are asymptotically not equivalent.