On the Eigenvalues and Eigenfunctions of the Sturm--Liouville Operator with a Singular Potential

In this paper we consider the Sturm--Liouville operators generated by the differential expression -y+q(x)y and by Dirichlet boundary conditions on the closed interval 0,]. Here q(x) is a distribution of first order,, i.e., q(x)dx L ₂0,]. Asymptotic formulas for the eigenvalues and eigenfunctions of such operators which depend on the smoothness degree of q(x) are obtained.