On a family of spectral problems for fourth-order ordinary differential operators

Academician Olga Ladyzhenskaya highlighted the significance of developing a fundamental system in the space of solenoidal functions for canonical domains such as cubes, squares, and related geometries. In principle, such a construction could be obtained through the solution of the spectral problem associated with the Stokes operator in these standard domains. In the present paper, we investigate a class of fourth-order ordinary differential operators defined on a finite interval. In the corresponding spectral problems naturally arise in the construction of a fundamental system in the space of solenoidal functions for a cubic domain. It is established that each problem from the considered family possesses a discrete countable spectrum, while the associated eigenfunctions form an orthogonal basis in an appropriate Sobolev functional space. Moreover, the basis property remains valid for the triple direct product of the constructed system. The analysis relies on functional-analytic methods, compact operator theory, and Hilbert–Schmidt spectral theory, providing a rigorous framework for the proposed construction and further developments. By additionally exploiting the properties of the curl operator, one can achieve the original objective without explicitly solving the spectral problem for the Stokes operator.