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

Zbl 0647.47052)],

*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.