Asymptotic behavior of eigenvalues and eigenfunctions in a non-self-adjoint problem for the Schrödinger operator with involution

In this paper, the spectral properties of non-self-adjoint boundary value problems for linear second-order differential equations with involution and variable complex-valued coefficients are investigated. The existence of linearly independent solutions is established, and the form of the general solution of linear second-order differential equations with involution and variable complex-valued coefficients is presented. Asymptotic formulas for linearly independent solutions of the equations under consideration are obtained. Using the asymptotics of linearly independent solutions, asymptotic formulas for eigenvalues and eigenfunctions of a non-self-adjoint spectral problem for a second-order differential equation with involution are proved. Estimates for the norms of derivatives of eigenfunctions are also established.