标题：On an open problem of Weidmann: essential spectra and square-integrable solutions
作者：Qi, Jiangang; Chen, Shaozhu
作者机构：[Qi, Jiangang; Chen, Shaozhu] Shandong Univ, Dept Math, Weihai 264209, Peoples R China.
通讯作者地址：[Qi, JG]Shandong Univ, Dept Math, Weihai 264209, Peoples R China.
来源：PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
摘要：In 1987, 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 an open problem of 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.