On the completeness of eigenfunctions of one $5$th-order differential operator

V. S. Rykhlov

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 68 (2022) no. 2, pp. 338-375

Résumé

In this paper, we fully solve the problem of the completeness of the eigenfunctions of an ordinary $5$th-order differential operator in the space of square-summable functions on the segment $[0,1]$ generated by the simplest differential expression $y^{(5)}$ and two-point two-term boundary conditions $\alpha_\nu y^{(\nu-1)}(0)+\beta_\nu y^{(\nu-1)}(1)=0$ and $\nu=\overline{1,5},$ under the main assumption $\alpha_\nu\neq 0,$ $\nu=\overline{1,5}$ or $\beta_\nu\neq 0,$ $\nu=\overline{1,5}$ (in this case, without loss of generality, we can assume that all $\alpha_\nu$ or all $\beta_\nu,$, respectively, are equal to one). The classical methods of studying completeness, which go back to well-known articles by M. V. Keldysh, A. P. Khromov, A. A. Shkalikov, and many others, are not applicable to the operator under consideration. These methods are based on “good” estimates for the spectral parameter of the used generating functions (“classical”) for the system of eigenfunctions and associated functions. In the case of a strong irregularity of the operator under consideration, these «classical» generating functions have too large rate of grows in the spectral parameter. To solve the problem of multiple completeness, we propose a new approach that uses a special parametric solution that generalizes «classical» generating functions. The main idea of this approach is to select the parameters of this special solution to construct generating functions that are no longer «classical» with suitable estimates in terms of the spectral parameter. Such a selection for the operator under consideration turned out to be possible, although rather nontrivial, which allowed us to follow the traditional scheme of proving the completeness of the system of eigenfunctions in the space of square-summable functions on the segment $[0,1].$