А Spectral Problem for a Differential Operator with Involution

The paper deals with a spectral problem for a differential operator with involution
\begin{equation*} L(y) = \alpha(x) y'(x) + y'(-x) + q(x) y(x) + r(x) y(-x). \end{equation*}
Here $\alpha$ is an absolutely continuous real function, while $q$ and $r$ are complex-valued summable functions. Two methods for studying the spectral properties of such an operator are proposed, which can be used to solve more general problems. The first method is based on the artificial construction of a dominant operator, information about the spectral properties of which can be obtained explicitly. After constructing such an operator, perturbation theory methods are used. The second method is based on reducing the spectral problem for the operator under consideration to a system of differential equations. The main results of the article are devoted to the study of the property of unconditional basis of eigenfunctions of a spectral problem generated by the operator $L$ with a regular boundary condition.