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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, forall $\l$ in an open interval $I\subset\bR$, the dimension of defect subspaces$\gN_\l(A)(=\Ker (A^*-\l))$ coincides with $d$, then every self-adjointextension $\wt A\supset A$ has no continuous spectrum in $I$ and the pointspectrum of $\wt A$ is nowhere dense in $I$. Application of this statement todifferential operators makes it possible to generalize the known results byWeidmann to the case of an ordinary differential expression with both singularendpoints and arbitrary equal deficiency indices of the minimal operator.Moreover, we show in the paper, that an old conjecture by Hartman and Wintneron the spectrum of a self-adjoint Sturm - Liouville operator is not valid.

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators