Vagabov A.I. The Task of Basis Property of Root Functions of Differential Sheaf of the 2nd Order With n-Fold Characteristics
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https://doi.org/10.15688/mpcm.jvolsu.2018.1.1
Abdulvagab Ismailovich Vagabov
Doctor of Physical and Mathematical Sciences, Professor,
Department of Mathematical Analysis,
Dagestan State University
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M. Gadzhieva St., 43-А, 367025 Makhachkala, Russian Federation
Abstract. The task under consideration is characterized by essential deviations from the viewpoint of widely famous regular (in Birkhoff-Tamarkin’s sense) spectral tasks (see [1; 3]). On the one hand, we have the n-multiplicity of each of the two characteristic roots of the differential expression. On the other hand, we adhere to the worst from the classical viewpoint case of disintegrating boundary conditions, when all but one of them are given at the left end and only one at the right end of the given interval.
The range of the studied problem is limited by imaginary eigenvalues that are equidistant from each other. Each eigenvalue is characterized by one proper and n – 1 functions attached to it. We construct the resolvent of the sheaf as a meromorphic function of the parameter λ.
We prove in the main theorem that the total residue with respect to a parameter from the resolvent applied to a 2n – 1 time differentiable function (vanishing together with the derivatives at the ends of the interval under consideration) is equal to this function. This residue, as it is well known, represents a Fourier series with respect to the root functions of the original task.
Key words: fundamental solutions, function of Cauchy, function of Green, range.
The Task of Basis Property of Root Functions of Differential Sheaf of the 2nd Order With n-Fold Characteristics by Vagabov A.I. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 21 No. 1 2018, pp. 5-10
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