Abstract
This paper considers a generalized spectral problem for a fourth-order differential operator. The primary goal of the research is to analyze the spectral properties of the operator arising in boundary value problems for the Stokes and Navier-Stokes equations, as well as to utilize the obtained eigenfunctions to construct a fundamental system in the space of solenoidal functions. The work combines theoretical analysis with practical applications, making it relevant for numerical modeling of hydrodynamic processes. The main methodology is based on the method of separation of variables and the use of curl operators for different domain dimensions. In particular, the paper proposes approaches to introducing curl operators for the three- and four-dimensional cases, which generalize the problem formulation. The key results include proving the existence and distribution of eigenvalues, as well as constructing an orthonormal basis in functional spaces. This study contributes to the development of spectral analysis of high-order operators and can be useful for developing efficient algorithms for solving hydrodynamic problems. The practical significance of the results lies in their application to numerical modeling of fluid flows in various fields of science and engineering.