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Let $A$ be a symmetric operator. By using the method of boundary triplets weparameterize in terms of a Nevanlinna parameter $\tau$ all exit spaceextensions $\wt A=\wt A^*$ of $A$ with the discrete spectrum $\s(\wt A)$ andcharacterize the Shtraus family of $\wt A$ in terms of abstract boundaryconditions. Next we apply these results to the eigenvalue problem for the$2r$-th order differential equation $l[y]= \l \D(x)y$ on an interval $[a,b), \;-\infty <a<b\leq \infty,$ subject to $\l$-depending separated boundaryconditions with entire operator-functions $C_0(\l) $ and $C_1(\l)$, which forma Nevanlinna pair $(C_0,C_1)$. The weight $\D(x)$ is nonnegative and may vanishon some intervals $(\a,\b)\subset \cI$. We show that in the case when theminimal operator of the equation has the discrete spectrum (in particular, inthe case of the quasiregular equation) the set of eigenvalues of the eigenvalueproblem is an infinite subset of $\bR$ without finite limit points and eachfunction $y\in\LI$ admits the eigenfunction expansion $y(x)=\sum_{k=1}^\inftyy_k (x)$ converging in $\LI$. Moreover, we give an explicit method forcalculation of eigenfunctions $y_k$ in this expansion and specify boundaryconditions on $y$ implying the uniform convergence of the eigenfunctionexpansion of y. These results develop the known ones obtained for the case ofthe positive weight $\D$ and the more restrictive class of $\l$-dependingboundary conditions.

On eigenfunction expansions of differential equations with degenerating weight