On Eigenfunction Expansions for a Class of Irregular Quadratic Pencils of Second-Order Differential Operators

A class of quadratic irregular pencils of the second order ordinary differential operators with constant coefficients is considered in this paper. It is assumed that the boundary conditions are splitting and the characteristic equation has positive roots. Both amounts of two-fold expansions in the series in terms of eigenfunctions of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found. For this class of irregular pencils, the question of a two-fold expansion in terms of eigenfunctions was not considered earlier.