Extension theory for symmetric operators

Extension theory and Nevanlinna R-functions

Probably the most interesting example zof a singular rank one perturbation is the perturbation of the boundary condition for a second order ordinary differential operator. This is one of the first mathematical problems where the extension theory plays an indispensable role. In 1909–1910 H. Weyl investigated in his famous papers [954, 955] the behavior of the solutions to the second order differential equation under variation of the boundary condition. He was also the first to ask the question: How does the spectrum change under such a perturbation? He proved that the absolutely continuous spectrum is invariant under such a perturbation. The question by H.Weyl concerning other types of spectrum has been investigated by F. Wolf [964], N. Aronszajn [97], and N.Aronszajn and W.F.Donoghue [99]. See also the paper by V.A. Javrian [493]. So it was H.Weyl who was the first to understand the importance of this class of perturbations from the mathematical point of view. The first mathematically rigorous investigation of singular perturbations of partial differential operator was carried out by F.A.Berezin and L.D.Faddeev [135]. These authors have shown that such perturbations can be described using the extension theory of symmetric operators. This paper was extremely important because it clarified the relation between partial differential operators with point interactions and Krein's formula describing the resolvents of all self–adjoint extensions of a given symmetric operator.