Operator models associated with singular perturbations (Q2777859)

Let \(S_0\) be a closed symmetric operator on a Hilbert space \(\mathfrak H_0\) with equal finite deficiency numbers \((d,d)\), and let \(M_0\) be the Weyl function of \(S_0\). For a monic \(d\times d\) matrix polynomial \(q\) and a selfadjoint \(d\times d\) matrix polynomial \(r\), the function \(M=r+q^\sharp M_0q\), where \(q^\sharp (\lambda)=q(\bar \lambda)^*\), is a generalized Nevanlinna function whose Nevanlinna kernel has \(dn\) negative squares. The authors construct an operator model to represent \(M\) as a Weyl function of a symmetric relation in a Pontryagin space \(\mathfrak H\) which contains \(\mathfrak H_0\) as a closed subspace. The construction is based on coupling methods involving models for sums and Schur complements of generalized Nevanlinna functions. The above model is used to obtain a (multivalued) operator interpretation of a singular finite rank perturbation of a selfadjoint operator.