Convergence and Approximation of the Sturm–Liouville Operators with Potentials-Distributions - Goryunov, A. S. (Ukrains’kyi Matematychnyi Zhurnal, 2015)
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.