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 $$ {Q}_n^2/{p}_n $$ 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.