On eigenvalues of self-adjoint extensions for defect larger than one

Self-adjoint extensions of a symmetric operator are parametrised by Krein's formula, in which the $Q$-function interacts with another analytic function (the parameter). We obtain a characterisation of the eigenvalues, isolated or not, of a given self-adjoint extension in terms of these two functions. The setting is highly general, covering symmetric operators with arbitrary defect in a Hilbert or Pontryagin space. Of independent interest is our newly developed tool, the \emph{generalised value} of a generalised Nevanlinna function, for which we give both a function-theoretic and an operator-theoretic description.