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.
Similar content being viewed by others
References
A. Zettl, Sturm–Liouville Theory, American Mathematical Society, Providence (2005).
S. Albeverio, F. Gestezy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, New York (1988).
S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators, Cambridge Univ. Press, Cambridge (2000).
A. S. Goriunov and V. A. Mikhailets, “Regularization of singular Sturm–Liouville equations,” Meth. Funct. Anal. Topol., 16, No. 2, 120–130 (2010).
A. S. Goriunov, V. A. Mikhailets, and K. Pankrashkin, “Formally self-adjoint quasi-differential operators and boundary-value problems,” Electron. J. Different. Equat., No. 101, 1–16 (2013).
J. Eckhardt, F. Gesztesy, R. Nichols, and G. Teschl, “Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional coefficients,” Opusc. Math., 33, No. 3, 467–563 (2013).
A. Zettl, “Formally self-adjoint quasi-differential operators,” Rocky Mountain J. Math., 5, No. 3, 453–474 (1975).
W. N. Everitt and L. Markus, Boundary-Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-Differential Operators, American Mathematical Society, Providence, RI (1999).
A. S. Goriunov and V. A. Mikhailets, “Regularization of two-term differential equations with singular coefficients by quasiderivatives,” Ukr. Math. J., 63, No. 9, 1190–1205 (2011).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
A. Savchuk and A. Shkalikov, “Sturm–Liouville operators with singular potentials,” Math. Notes, 66, No. 5-6, 741–753 (1999).
A. S. Goriunov and V. A. Mikhailets, “Resolvent convergence of Sturm–Liouville operators with singular potentials,” Math. Notes, 87, No. 1-2, 287–292 (2010).
J. Yan and G. Shi, “Inequalities among eigenvalues of Sturm–Liouville problems with distribution potentials,” J. Math. Anal. Appl., 409, No. 1, 509–520 (2014).
T. I. Kodlyuk, V. A. Mikhailets, and N. V. Reva, “Limit theorems for one-dimensional boundary-value problems,” Ukr. Math. J., 65, No. 1, 77–90 (2013).
A. Yu. Levin, “Limit Transition for the nonsingular systems Ẋ = A n (t)X,” Dokl. Akad. Nauk SSSR, 176, No. 4, 774–777 (1967).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 5, pp. 602–610, May, 2015.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1107-z