On completeness conditions for system of root functions of differential operator on segment with integral conditions

Abstract

In the work we study the completeness conditions for the system of root functions (SRF) of the operator $L_U$ generated by the differential expression
$$l(y)=-y''+qy \quad (q\in L_1(0,1))$$
and the integral conditions
$$y^{(j-1)}(0)+(l(y),u_j)=0 \quad (u_j\in L_2(0,1),\ j=1,2)$$
in the space $H=L_2(0,1)$. We show that SRF of the operator $L_U$ is complete in its domain if there exist two rays in the upper half-plane such for all large $\lambda$ on these rays the characteristic determinant is bounded from below by the function $\lambda^{m}e^{-|\mathrm{Im}\,\lambda|}$, $\displaystyle{m\geq\frac{1}{2}}$. If the operator $L_U$ is densely defined, then to ensure the completeness of SRF in $H$, it is sufficient to have the mentioned estimate with an arbitrary $m\in \mathbb{R}$. Moreover, we obtain an integral representation for the characteristic determinant as the sine--transform of some function $A$, which is expressed via $u_1$, $u_2$ and the kernel of the transformation operator for the equation $l(y)=\lambda^2y$. Employing this representation, we find explicit (in terms of the functions $u_1,u_2$) completeness conditions for SRF of the operator $L_U$ in $H$ or $D(L_U)$.