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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 5, pp. 602–610, May, 2015.
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Horyunov, A.S. Convergence and Approximation of the Sturm–Liouville Operators with Potentials-Distributions. Ukr Math J 67, 680–689 (2015). https://doi.org/10.1007/s11253-015-1107-z
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DOI: https://doi.org/10.1007/s11253-015-1107-z