One-dimensional Perturbations, Asymptotic Expansions, and Spectral Gaps

Let S be a closed symmetric operator or relation with defect numbers (1, 1) and let A be a self-adjoint extension of S. The self-adjoint extensions A(tau), tau is an element of R boolean OR {infinity}, of S, when parametrized by means of Krein's formula, can be seen as one-dimensional (graph) perturbations of A. The spectral properties of the self-adjoint extension A(tau) of (the completely non-self-adjoint part of) S can be determined via the analytic properties of the Weyl function (Q-function) Q(tau) (z) corresponding to S and A(tau), and conversely. In order to study the limiting properties of these functions at spectral points, local analogs of the Kac-Donoghue classes of Nevanlinna functions are introduced, giving rise to asymptotic expansions at real points. In the case where the self-adjoint extension A has a (maximal) gap in its spectrum, all the perturbations A(tau) have the same gap in their spectrum with the possible exception of an isolated eigenvalue lambda(tau), tau is an element of R boolean OR {infinity}. By means of the Weyl function Q(tau) (z) the (analytic) properties of this eigenvalue are established.