Boundary-Value Problems with Birkhoff Regular but not Strongly Regular Conditions for a Second-Order Differential Operator

Abstract

We study the self-adjoint problems whose operators split in the invariant subspaces induced by the involution operator Iy(x) = y(1− x). We construct nonself-adjoint perturbations of these problems that are Birkhoff regular but not strongly regular and, for some values of the coefficients of the boundary conditions transform into nonspectral problems in Dunford’s sense. We study the spectral properties of operators corresponding to these perturbations and, in particular, determine the eigenvalues and root functions and analyze the completeness and basis property of the system of root functions. We find the families of boundary conditions that generate essentially nonself-adjoint problems and contain the nonlocal Samarskii–Ionkin conditions.