ASYMPTOTIC BEHAVIOR OF THE VECTOR-FUNCTIONS OF OPERATORS APPROXIMATING THE DIFFERENTIAL EQUATIONS WITH δ-SHAPED COEFFICIENTS
Abstract
The equations can be written as L0u= − u∆+a(ε) δu =f which appear in different applications and are studied intensively. In this equation, δu is not determined in the classical theory of generalized functions, so one of the main objectives is to give meaning to the expression on the left-hand side of the equation, that is, it is an actual construction of the operator that corresponds to a given formal expression. This is achieved by special approximations of multiplication of the operator by the δ-function. To study equations with δ-shaped coefficients we have applied the approach, the main steps of which are: constructing the approximations of the considered expressions with operators of finite rank; finding the explicit form approximating a resolvent family; determining a resolvent limit and allocating resonance cases; describing the spectrum of the constructed limit of operators; studying the behavior of the eigenvalues of approximating operators. The purpose of this work is to find the asymptotic behavior of vector-functions for approximations, built in [2]. Thus, the main result of this work is the construction of the asymptotic behavior of the vector-functions in different cases of resonance.
References
1. Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H., Exner P. Solvable models in quantum mechanics. Berlin, Springer, 1988. 458 p. Doi: 10.1007/978-3-642-88201-2
2. Kot M. G. About resolvent convergence of operator approximating systems of equations with δ-shaped coefficints. Vestnik BGU. Seriya 1, Fizika. Matematika. Informatika = Vestnik BSU. Series 1: Physics. Mathematics. Informatics, 2015, no. 3, pp.111–117 (in Russian).
3. Antonevich A. B., Romanchuk T. A. Approximation operators with delta -shaped coefficints. Aktualnye problemy matematiki: sbornik nauchnyh trudov [Actual problems of mathematics: the collection of scientific papers]. Grodno, Grodno State University, 2008, pp. 11–28 (in Russian).
4. Antonevich A. B., Romanchuk T. A. Equations with delta-shaped coefficients: method of finite-dimensional approximations. Saarbrücken, Laplambert, 2012. 148 p. (in Russian).
5. Kot M. G. Asymptotics of the eigenvalues of approximating differential equations with δ-different coefficients. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika = Journal of the Belarusian State University. Mathematics and Informatics, 2017, no. 1, pp. 3–10 (in Russian).
6. Kaschenko I. S. The asymptotic expansion of the solutions of equations: the method of guidance. Yaroslavl, Yaroslavl State University, 2011. 44 p. (in Russian).
7. Vasil’ev V. A. Asymptotic exponential integrals, Newton’s diagram and classification of minimum points. Functional Analysis and Its Applications, 1977, vol. 11, no. 3, pp. 163–172. Doi: 10.1007/bf01079460
8. Zabreiko P. P, Krivko-Kras’ko A. V. Newton diagrams and algebraic curves. Trudy Instituta matematiki = Proceedings of the Institute of Mathematics, 2014, vol. 22, no. 2, pp. 32–45 (in Russian).
9. Zabreiko P. P, Krivko-Kras’ko A. V. Newton diagrams and algebraic curves II. Trudy Instituta matematiki = Proceedings of the Institute of Mathematics, 2015, vol. 23, no. 1, pp. 64–75 (in Russian).
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