Multiple completeness of the root functions of some nonregular pencils of the third order differential operators.

Three concrete examples of strongly nonregular polynomial pencils of ordinary differential operators of the third order generated on $[0, 1]$ by a linear differential expression with constant coefficients, polynomially depending on spectral parameter $\lambda$, and by two-point not semisplitting boundary conditions are considered, namely:

$1)$ the pencil $L^{1}_{0}(\lambda)$ of the form

$y'''−\lambda y''+ \lambda^{2}y' − \lambda^{3}y=0$,

$y(0)+y(1)=y'(0)+iy'(1)=y''(0)−y''(1)=0$;

$2)$ the pencil $L^{2}_{0}(λ)$ of the form

$y'''−(1+i)\lambda y''+(2+i)\lambda^{2}y'−2\lambda^{3}y = 0$,

$y(0)−5y(1)=y'(0) − (2 + 6i)y'(1) = y''(0) + 10y''(1) = 0$;

$3)$ the pencil $L^{3}_{0}(\lambda)$ of the form

$y'''−(1−i)\lambda y''+(1+2i)\lambda^{2}y'−(1+3i)\lambda^{3}y=0$,

$y(0)+5y(1)=y'(0)+(1+2i)y'(1)=y''(0)−(3+i)y''(1)+(6−3i)\lambda y'(1)=0$.

Based on general theorems on multiple completeness of the root functions, obtained earlier by the author, multiple completeness of the root functions of the pencils in the space $L^{2}[0, 1]$ is investigated. It is found that in spite on a similar form of pencils of these examples, the multiplicity of the completeness of the root functions is completely different.

For the first example the root functions are one-fold complete in the space $L^{2}[0, 1]$ with a possible finite defect and two-fold incomplete with infinite defect.For the second example the root functions are two-fold complete in the space $L^{2}[0, 1]$ with a possible finite defect and three-fold incomplete with infinite defect.

And for the third example the root functions are three-fold complete in the space$L^{2}[0, 1]$.

Appropriate sets of vector-functions, orthogonal to derivative chains of corresponding multiplicity of the considered pencils, built by the root functions of the considered pencils, are constructed.

The method of investigating of multiple completeness of the root functions is to use a special solution of the basic differential equation depending on an arbitrary parameter vector. An important role is played by lemma on the characteristic polygons of the special solutions when as vector-parameters are taken vectors constructed on the basis of the coefficients of the boundary conditions and the roots of the characteristic polynomial.

Keywords: pencil of ordinary differential operators, pencil of the third order, nonregular pencil, root functions, eigen- and associated functions, multiple completeness, constant coefficients of differential expression, not semisplitting boundary conditions, two-point boundary conditions.