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

Abstract

We study the operators $L_n y = −(p_n y′)′+q_n y, n ∈ ℤ_{+}$, given on a finite interval with various boundary conditions. It is assumed that the function $q_n$ is a derivative (in a sense of distributions) of $Q_n$ and $1/p_n , Q_n /p_n$, and $Q^2_n/p_n$ are integrable complex-valued functions. The sufficient conditions for the uniform convergence of Green functions $G_n$ of the operators $L_n$ on the square as $n → ∞$ to $G_0$ are established. It is proved that every $G_0$ is the limit of Green functions of the operators $L_n$ with smooth coefficients. If $p_0 > 0$ and $Q_0(t) ∈ ℝ$, then they can be chosen so that $p_n > 0$ and $q_n$ are real-valued and have compact supports.