One­-dimensional graph perturbations of selfadjoint relations

Let A be a selfadjoint operator (or a selfadjoint relation) in a Hilbert space h, let Z be a one­dimensional subspace of h^2 such that A∩Z = {0, 0} and define S = A∩Z*. Then S is a closed, symmetric operator (or relation) with defect numbers (1, 1) and, conversely, each such S and a selfadjoint extension A are related in this way. This allows us to interpret the selfadjoint extensions of S in h as one-­dimensional graph perturbations of A. If Z = span {φ, ψ}, then the function Q(l) = l[φ, φ] + [(A - l)^-1 (lφ - ψ), ¯lφ - ψ], generated by A and the pair {φ, ψ}, is a Q-­function of S = A∩Z* and A. It belongs to the class N of Nevanlinna functions and essentially determines S and A. Calculation of the corresponding resolvent operators in the perturbation formula leads to Kreĭn's description of (the resolvents of) the selfadjoint extensions of S. The class N of Nevanlinna functions has subclasses N1 ⊃ N0 ⊃ N-1 ⊃ N-2, each defined in terms of function-­theoretic properties. We characterize the Q-­functions belonging to each of these classes in terms of the pair {φ, ψ}. If Q(l) belongs to the subclass Nk, k = 1, 0, -1, -2, then all but one of the selfadjoint extensions of S have a Q-­function in the same class, while the exceptional extension has a Q-function in N N1. In particular, if S is semi-bounded, the exceptional selfadjoint extension is precisely the Friedrichs extension. We consider our perturbation formula in the case where the Q-­function Q(l) belongs to the subclass Nk, k = 1, 0, -1, -2, or if it is an exceptional function associated with this subclass. The resulting perturbation formulas are made explicit for the case that A or its orthogonal operator part is the multiplication operator in a Hilbert space L^2(dρ).