Abstract:
In this paper we studied the spectral problem for an ordinary second order differential equation on a finite interval with a discontinuous coefficient of the highest derivative. At the ends of the segment the boundary conditions of the first kind are given. We found eigenvalues with their asymptotic behavior as the roots of the transcendental equation. The system of eigenfunctions is the trigonometric sine on one half of the segment, and the hyperbolic sine on the other. The system of eigenfunctions is not orthogonal in the space of square integrable functions. The corresponding biorthogonal system of functions was built as a solution to the dual problem. In the proof of the completeness of the biorthogonal system we used well known Keldysh theorem about the completeness of the eigenfunctions system of a nonselfadjoint operator.
Keywords:
eigenvalues, eigenfunctions, adjoint problem, complete system of functions.
Citation:
A. A. Gimaltdinova, K. V. Kurman, “On the completeness of a pair of biorthogonally conjugated systems of functions”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:1 (2015), 7–18
\Bibitem{GimKur15}
\by A.~A.~Gimaltdinova, K.~V.~Kurman
\paper On the completeness of a pair of biorthogonally conjugated systems of functions
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2015
\vol 19
\issue 1
\pages 7--18
\mathnet{http://mi.mathnet.ru/vsgtu1385}
\crossref{https://doi.org/10.14498/vsgtu1385}
\zmath{https://zbmath.org/?q=an:06968945}
\elib{https://elibrary.ru/item.asp?id=23681739}
Linking options:
https://www.mathnet.ru/eng/vsgtu1385
https://www.mathnet.ru/eng/vsgtu/v219/i1/p7
This publication is cited in the following 2 articles:
A. Gimaltdinova, “The Dirichlet problem for an equation of mixed type with two internal lines of type change”, Lobachevskii J. Math., 41:11, SI (2020), 2155–2167
A. A. Gimaltdinova, “Second boundary-value problem for the Lavrent'ev–Bitsadze equation in a rectangular domain with two degeneration lines”, J. Math. Sci. (N. Y.), 236:6 (2019), 579–593
Citation in format AMSBIB
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