Convergence and Approximation of the Sturm–Liouville Operators with Potentials-Distributions

We study the operators Lny = −(pny′)′+qny, n ∈ ℤ+, given on a finite interval with various boundary conditions. It is assumed that the function qn is a derivative (in a sense of distributions) of Qn and 1/pn, Qn/pn, and Qn2/pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {Q}_n^2/{p}_n $$\end{document} are integrable complex-valued functions. The sufficient conditions for the uniform convergence of Green functions Gn of the operators Ln on the square as n → ∞ to G0 are established. It is proved that every G0 is the limit of Green functions of the operators Ln with smooth coefficients. If p0> 0 and Q0(t) ∈ ℝ, then they can be chosen so that pn> 0 and qn are real-valued and have compact supports.