On completeness conditions for system of root functions of differential operator on segment with integral conditions
Kh. K. Ishkina,
B. E. Kanguzhinb a Ufa University of Science and Technology, Zaki Validi str. 32, 450074, Ufa, Russia
b al-Farabi Kazakh National University, al-Farabi av. 71, A15E3B4, Almaty, Kazakhstan
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
$$l(y)=-y''+qy (q\in L_1(0,1))$$
and the integral conditions
$$y^{(j-1)}(0)+(l(y),u_j)=0 (u_j\in L_2(0,1),\ j=1,2)$$
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
$\lambda^{m}e^{-|\mathrm{Im} \lambda|}$,
$m\geq\frac{1}{2}$. 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
$D(L_U)$.
Keywords:
differential operator with integral boundary conditions, spectrum, asymptotics.
UDC:
517.984 +
517.928
MSC: 34L10,
47B28 Received: 21.08.2025
Fulltext:
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