On $2$-fold completeness of the eigenfunctions for the strongly irregular quadratic pencil of differential operators of second order

A class of strongly irregular pencils of ordinary differential operators of second order with constant coefficients is considered. The roots of the characteristic equation of the pencils from this class are supposed to lie on a straight line coming through the origin and on the both side of the origin. Exact interval on which the system of eigenfunctions is $2$-fold complete in the space of square summable functions is finded.