Jakob Reiffenstein: Krein’s formula, eigenvalues, and generalized values of Nevanlinna functions

Abstract: The self-adjoint extensions of a symmetric operator S with defect n > 1 are parametrized by Krein’s formula
\(\mathcal{P}_\mathcal{H}(\tilde{A}-\lambda)^{-1}|_\mathcal{H}=(A-\lambda)^{-1}-\gamma(\lambda)(m(\lambda)+\tau(\lambda))^{-1}\gamma(\overline{\lambda})^*\)
Here A is a “default” extension with defect family γ(λ) and Weyl function m(λ). The function τ (λ) is seen as the parameter, and just like m(λ) it is an n × n matrix-valued Herglotz-Nevanlinna function. In this talk I will present a complete characterization of the eigenvalues of A, including those that are embedded in the essential spectrum, in terms of m(λ) and τ (λ). I will explain why this requires not only the use of generalized zeros and poles, but also of a new concept called generalized values. Based on joint work with A. Luger.