On some extensions of the A-model

TL;DR

This paper explores extensions of the A-model for finite rank singular perturbations using boundary relations, providing new constructions under specific Hilbert space decompositions.

Contribution

It introduces novel extension methods for the A-model in the context of boundary relations and Hilbert space decompositions.

Findings

Constructed nontrivial extensions in the A-model for symmetric restrictions.

Utilized boundary relations to analyze finite rank singular perturbations.

Extended the theoretical framework for the A-model in Hilbert space settings.

Abstract

The A-model for finite rank singular perturbations of class $\mathfrak{H}_{-m-2}\smallsetminus\mathfrak{H}_{-m-1}$, $m\in\mathbb{N}$, is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces $(\mathfrak{H}_n)_{n\in\mathbb{Z}}$ admit an orthogonal decomposition $\mathfrak{H}^-_n\oplus\mathfrak{H}^+_n$, with the corresponding projections satisfying $P^\pm_{n+1}\subseteq P^\pm_n$, nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.