Spectral Problem for a Differential Operator with Involution

We consider the spectral problem for a differential operator with involution $$L(y) = \alpha(x) y'(x) + y'(-x) + q(x) y(x) + r(x) y(-x).$$ Here $$\alpha$$ is an absolutely continuous real-valued function, while $$q$$ and $$r$$ are complex-valued integrable functions. We propose two methods for studying the spectral properties of such an operator, which can also be applied to more general problems. The first method is based on the construction of an auxiliary dominating operator whose spectral properties can be described explicitly. After constructing such an operator, we apply methods of perturbation theory. The second method is based on reducing the spectral problem for the operator to a system of differential equations. The main results of the paper provide sufficient conditions for the unconditional basis property of the eigenfunctions of the spectral problem associated with the operator $$L$$ under a regular boundary condition.