UDC

517.927.25

Authors

Konechnaya Natal’ya Nikolaevna
Institute of Mathematics, Information and Space Technologies,
Northern (Arctic) Federal University named after M.V. Lomonosov Uritskiy str., 68, Arkhangelsk, 163002, Russian Federation;
e-mail: 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;
e-mail: 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;
e-mail: tagirova_rena@mail.ru

Abstract

One 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 Sturm-Liouville operator generated by the expression –yʺ(x) + q(x)y(x) and self-adjoint boundary conditions in space L₂[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 Sturm-Liouville 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 Sturm-Liouville operator with the first order distribution potential as an operator generated by the quasi-differential 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 self-adjoint boundary conditions in space L₂[–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.

Keywords

quasi-differential operators, Sturm-Liouville operator with δ-potential, asymptotics of the eigenvalues, regularized trace of the operator

References

1. Savchuk A.M., Shkalikov A.A. Operatory Shturma–Liuvillya s singulyarnymi po-tentsialami [Sturm- Liouville Operators with Singular Potentials]. Matematicheskie zametki [Mathematical Notes], 1999, vol. 66, no. 3, pp. 847–912. 
2. Savchuk A.M., Shkalikov A.A. Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami [Sturm- Liouville Operators with Distributional Potentials]. Trudy moskovskogo matematicheskogo obshchestva, 2003, vol. 64, pp. 159–212. 
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4. Savchuk A.M. Regulyarizovannyy sled pervogo poryadka operatora Shturma–Liuvillya s δ-potentsialom [Regularized Trace of the First-Order Sturm-Liouville Operator with δ-Potential]. Uspekhi matematicheskikh nauk [Russian Mathematical Surveys], 2000, vol. 55, no. 6, pp. 155–156. 
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