  Let $\gH$ be a Hilbert space and let $A$ be a simple symmetric operator in
$\gH$ with equal deficiency indices $d:=n_\pm(A)<\infty$. We show that if, for
all $\l$ in an open interval $I\subset\bR$, the dimension of defect subspaces
$\gN_\l(A)(=\Ker (A^*-\l))$ coincides with $d$, then every self-adjoint
extension $\wt A\supset A$ has no continuous spectrum in $I$ and the point
spectrum of $\wt A$ is nowhere dense in $I$. Application of this statement to
differential operators makes it possible to generalize the known results by
Weidmann to the case of an ordinary differential expression with both singular
endpoints and arbitrary equal deficiency indices of the minimal operator.
Moreover, we show in the paper, that an old conjecture by Hartman and Wintner
on the spectrum of a self-adjoint Sturm - Liouville operator is not valid.
