Sturm-liouville operators with singular potentials

This paper deals with Sturm-Liouville operators generated on a finite interval and on the whole axis by the differential expressionl(y)=−y"+q(x)y, whereq(x) is a distribution of first order, such that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\smallint q(\varepsilon )d\varepsilon \in L_{{\text{2,loc}}} $$ \end{document}. The minimal and maximal operators corresponding to potentials of this type on a finite interval are constructed. All self-adjoint extensions of the minimal operator are described and the asymptotics of the eigenvalues of these extensions is found. It is proved that the constructed operator coincides with the norm resolvent limit of the Sturm-Liouville operators generated by smooth potentialsqn, provided that the condition\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\smallint |\smallint (q_n - q)d\varepsilon |^{\text{2}} dx \to 0$$ \end{document} holds. The convergence of the spectra of these operators to the spectrum of the limit operator is also proved. Similar results are obtained in the case of the whole axis.