Authorsâ€™ abstract: In [Spectral theory of ordinary differential operators. Lecture Notes in Mathematics 1258. Berlin etc.: Springer-Verlag (1987;

Zbl 0647.47052)], {\it J. Weidmann} proved that, for a symmetric differential operator $\tau $ and a real $\lambda $, if there exist fewer square-integrable solutions of $(\tau - \lambda )y = 0$ than needed and if there is a self-adjoint extension of $\tau $ such that $\lambda $ is not its eigenvalue, then $\lambda $ belongs to the essential spectrum of $\tau $. However, he posed as open problem whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.