Arctic Environmental Research
A peerreviewed openaccess journal
eISSN 26587173 DOI:10.17238/issn25418416
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Section: Physics. Mathematics. Informatics UDC517.927.25AuthorsKonechnaya Natal’ya NikolaevnaInstitute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov Uritskiy str., 68, Arkhangelsk, 163002, Russian Federation; email: n.konechnaya@narfu.ru Safonova Tat’yana Anatol’evna Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation; email: t.Safonova@narfu.ru Tagirova Rena Nasirovna Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation; email: tagirova_rena@mail.ru AbstractOne of the interesting problems of the spectral theory of operators is the study of asymptotic behavior of the distribution function for the large values of the spectral parameter λ . A particular case of this problem is to study the asymptotics of the eigenvalues, eigenfunctions depending on the properties of the coefficients of differential expressions and to obtain the formulas of regularized trace for the corresponding operators. The significant results for the differential SturmLiouville operator generated by the expression –yʺ(x) + q(x)y(x) and selfadjoint boundary conditions in space L_{2}[a, b], with a continuously differentiable potential were obtained by I.M. Gelfand, B.M. Levitan in 1953. More recently, A.A. Shkalikov, A.M. Savchuk in their papers first obtained the asymptotics of the eigenvalues, eigenfunctions and the formula of the regularized trace for the SturmLiouville operators on a finite interval with singular potentials that are not locally integrable functions, and Dirichlet boundary conditions. At the same time, the definition of the SturmLiouville operator with the first order distribution potential as an operator generated by the quasidifferential second order expression with locally summable coefficients first presented in the papers of A.A. Shkalikov and A.M. Savchuk was applied. This approach allowed us to investigate in this paper the asymptotic behavior of the eigenvalues and obtain the formulas of the first order regularized trace for the operators generated by the expression –yʺ(x) + hδ(x)y(x), where δ(x) – δ – Dirac delta function, h ∈R , and some selfadjoint boundary conditions in space L_{2}[–1, 1], namely by the conditions of the type: i) y(–1) = y(1) = 0; ii) y^{[1}](–1) = y^{[1]}(1) = 0; iii) y(–1) = y^{[1]}(1) = 0; iv) y(–1) = y(1), y^{[1]}(–1) = y^{[1]}(1). The corresponding transcendental equations were established to get the asymptotics of the eigenvalues of these operators. Further analysis of the equations allows us to obtain the formulas of the first order regularized trace for these operators.Keywordsquasidifferential operators, SturmLiouville operator with δpotential, asymptotics of the eigenvalues, regularized trace of the operatorReferences
